diff --git a/src/category-theory/augmented-simplex-category.lagda.md b/src/category-theory/augmented-simplex-category.lagda.md
index fd816bc592..9c59e0e7b2 100644
--- a/src/category-theory/augmented-simplex-category.lagda.md
+++ b/src/category-theory/augmented-simplex-category.lagda.md
@@ -16,6 +16,7 @@ open import elementary-number-theory.natural-numbers
 open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 
 open import order-theory.order-preserving-maps-posets
@@ -81,19 +82,19 @@ associative-comp-hom-augmented-simplex-Category {n} {m} {r} {s} =
     ( Fin-Poset r)
     ( Fin-Poset s)
 
-inv-associative-comp-hom-augmented-simplex-Category :
+involutive-eq-associative-comp-hom-augmented-simplex-Category :
   {n m r s : obj-augmented-simplex-Category}
   (h : hom-augmented-simplex-Category r s)
   (g : hom-augmented-simplex-Category m r)
   (f : hom-augmented-simplex-Category n m) →
-  comp-hom-augmented-simplex-Category {n} {r} {s}
-    ( h)
-    ( comp-hom-augmented-simplex-Category {n} {m} {r} g f) ＝
   comp-hom-augmented-simplex-Category {n} {m} {s}
     ( comp-hom-augmented-simplex-Category {m} {r} {s} h g)
-    ( f)
-inv-associative-comp-hom-augmented-simplex-Category {n} {m} {r} {s} =
-  inv-associative-comp-hom-Poset
+    ( f) ＝ⁱ
+  comp-hom-augmented-simplex-Category {n} {r} {s}
+    ( h)
+    ( comp-hom-augmented-simplex-Category {n} {m} {r} g f)
+involutive-eq-associative-comp-hom-augmented-simplex-Category {n} {m} {r} {s} =
+  involutive-eq-associative-comp-hom-Poset
     ( Fin-Poset n)
     ( Fin-Poset m)
     ( Fin-Poset r)
@@ -104,16 +105,9 @@ associative-composition-operation-augmented-simplex-Category :
     hom-set-augmented-simplex-Category
 pr1 associative-composition-operation-augmented-simplex-Category {n} {m} {r} =
   comp-hom-augmented-simplex-Category {n} {m} {r}
-pr1
-  ( pr2
-      associative-composition-operation-augmented-simplex-Category
-        { n} {m} {r} {s} h g f) =
-  associative-comp-hom-augmented-simplex-Category {n} {m} {r} {s} h g f
-pr2
-  ( pr2
-      associative-composition-operation-augmented-simplex-Category
-        { n} {m} {r} {s} h g f) =
-  inv-associative-comp-hom-augmented-simplex-Category {n} {m} {r} {s} h g f
+pr2 associative-composition-operation-augmented-simplex-Category
+  { n} {m} {r} {s} =
+  involutive-eq-associative-comp-hom-augmented-simplex-Category {n} {m} {r} {s}
 
 id-hom-augmented-simplex-Category :
   (n : obj-augmented-simplex-Category) → hom-augmented-simplex-Category n n
diff --git a/src/category-theory/categories.lagda.md b/src/category-theory/categories.lagda.md
index 79427a3da2..e015106c69 100644
--- a/src/category-theory/categories.lagda.md
+++ b/src/category-theory/categories.lagda.md
@@ -20,6 +20,7 @@ open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.surjective-maps
 open import foundation.universe-levels
 ```
@@ -28,7 +29,7 @@ open import foundation.universe-levels
 
 ## Idea
 
-A **category** in Homotopy Type Theory is a
+A {{#concept "category" Agda=Category}} in Homotopy Type Theory is a
 [precategory](category-theory.precategories.md) for which the
 [identifications](foundation-core.identity-types.md) between the objects are the
 [isomorphisms](category-theory.isomorphisms-in-precategories.md). More
@@ -105,15 +106,15 @@ module _
   associative-comp-hom-Category =
     associative-comp-hom-Precategory precategory-Category
 
-  inv-associative-comp-hom-Category :
+  involutive-eq-associative-comp-hom-Category :
     {x y z w : obj-Category}
     (h : hom-Category z w)
     (g : hom-Category y z)
     (f : hom-Category x y) →
-    comp-hom-Category h (comp-hom-Category g f) ＝
-    comp-hom-Category (comp-hom-Category h g) f
-  inv-associative-comp-hom-Category =
-    inv-associative-comp-hom-Precategory precategory-Category
+    comp-hom-Category (comp-hom-Category h g) f ＝ⁱ
+    comp-hom-Category h (comp-hom-Category g f)
+  involutive-eq-associative-comp-hom-Category =
+    involutive-eq-associative-comp-hom-Precategory precategory-Category
 
   associative-composition-operation-Category :
     associative-composition-operation-binary-family-Set hom-set-Category
diff --git a/src/category-theory/composition-operations-on-binary-families-of-sets.lagda.md b/src/category-theory/composition-operations-on-binary-families-of-sets.lagda.md
index 7e4f2bee3f..d4e50dfdf4 100644
--- a/src/category-theory/composition-operations-on-binary-families-of-sets.lagda.md
+++ b/src/category-theory/composition-operations-on-binary-families-of-sets.lagda.md
@@ -14,6 +14,7 @@ open import foundation.identity-types
 open import foundation.iterated-dependent-product-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
 open import foundation.universe-levels
 ```
@@ -22,22 +23,24 @@ open import foundation.universe-levels
 
 ## Idea
 
-Given a type `A`, a **composition operation on a binary family of sets**
-`hom : A → A → Set ` is a map
+Given a type `A`, a
+{{#concept "composition operation" Disambiguation="on binary families of sets" Agda=composition-operation-binary-family-Set}}
+on a binary family of [sets](foundation-core.sets.md) `hom : A → A → Set` is a
+map
 
 ```text
-  hom y z → hom x y → hom x z
+  _∘_ : hom y z → hom x y → hom x z
 ```
 
 for every triple of elements `x y z : A`.
 
 For such operations, we can consider
-[properties](foundation-core.propositions.md) such as **associativity** and
-**unitality**.
+[properties](foundation-core.propositions.md) such as _associativity_ and
+_unitality_.
 
 ## Definitions
 
-### Composition operations in binary families of sets
+### Composition operations on binary families of sets
 
 ```agda
 module _
@@ -50,14 +53,30 @@ module _
     type-Set (hom-set y z) → type-Set (hom-set x y) → type-Set (hom-set x z)
 ```
 
-### Associative composition operations in binary families of sets
+### Associative composition operations on binary families of sets
 
-We give a slightly nonstandard definition of associativity, requiring an
-associativity witness in each direction. This is of course redundant as `inv` is
-a [fibered involution](foundation.fibered-involutions.md) on
-[identity types](foundation-core.identity-types.md). However, by recording both
-directions we maintain a definitional double inverse law which is practical in
-defining the [opposite category](category-theory.opposite-categories.md).
+A composition operation
+
+```text
+  _∘_ : hom y z → hom x y → hom x z
+```
+
+on a binary family of sets of morphisms is called
+{{#concept "associative" Disambiguation="composition operation on a binary family of sets" Agda=is-associative-composition-operation-binary-family-Set}}
+if, for every triple of composable morphisms we have
+
+```text
+  (h ∘ g) ∘ f ＝ h ∘ (g ∘ f).
+```
+
+We give a slightly nonstandard definition of associativity using the
+[strictly involutive identity types](foundation.strictly-involutive-identity-types.md)
+rather than the standard [identity types](foundation-core.identity-types.md).
+This is because, while the strictly involutive identity types are always
+[equivalent](foundation-core.equivalences.md) to the standard ones, they satisfy
+the strict computation rule `inv (inv p) ≐ p` which is practical in defining the
+[opposite category](category-theory.opposite-categories.md), as this also makes
+the opposite construction strictly involutive: `(𝒞ᵒᵖ)ᵒᵖ ≐ 𝒞`.
 
 ```agda
 module _
@@ -71,8 +90,7 @@ module _
     (h : type-Set (hom-set z w))
     (g : type-Set (hom-set y z))
     (f : type-Set (hom-set x y)) →
-    ( comp-hom (comp-hom h g) f ＝ comp-hom h (comp-hom g f)) ×
-    ( comp-hom h (comp-hom g f) ＝ comp-hom (comp-hom h g) f)
+    ( comp-hom (comp-hom h g) f ＝ⁱ comp-hom h (comp-hom g f))
 
   associative-composition-operation-binary-family-Set : UU (l1 ⊔ l2)
   associative-composition-operation-binary-family-Set =
@@ -88,6 +106,19 @@ module _
     composition-operation-binary-family-Set hom-set
   comp-hom-associative-composition-operation-binary-family-Set = pr1 H
 
+  involutive-eq-associative-composition-operation-binary-family-Set :
+    {x y z w : A}
+    (h : type-Set (hom-set z w))
+    (g : type-Set (hom-set y z))
+    (f : type-Set (hom-set x y)) →
+    ( comp-hom-associative-composition-operation-binary-family-Set
+      ( comp-hom-associative-composition-operation-binary-family-Set h g)
+      ( f)) ＝ⁱ
+    ( comp-hom-associative-composition-operation-binary-family-Set
+      ( h)
+      ( comp-hom-associative-composition-operation-binary-family-Set g f))
+  involutive-eq-associative-composition-operation-binary-family-Set = pr2 H
+
   witness-associative-composition-operation-binary-family-Set :
     {x y z w : A}
     (h : type-Set (hom-set z w))
@@ -98,7 +129,8 @@ module _
     ( comp-hom-associative-composition-operation-binary-family-Set
       ( h) (comp-hom-associative-composition-operation-binary-family-Set g f))
   witness-associative-composition-operation-binary-family-Set h g f =
-    pr1 (pr2 H h g f)
+    eq-involutive-eq
+      ( involutive-eq-associative-composition-operation-binary-family-Set h g f)
 
   inv-witness-associative-composition-operation-binary-family-Set :
     {x y z w : A}
@@ -110,50 +142,32 @@ module _
     ( comp-hom-associative-composition-operation-binary-family-Set
       ( comp-hom-associative-composition-operation-binary-family-Set h g) (f))
   inv-witness-associative-composition-operation-binary-family-Set h g f =
-    pr2 (pr2 H h g f)
+    eq-involutive-eq
+      ( invⁱ
+        ( involutive-eq-associative-composition-operation-binary-family-Set
+          ( h)
+          ( g)
+          ( f)))
 ```
 
-```agda
-module _
-  {l1 l2 : Level} {A : UU l1}
-  (hom-set : A → A → Set l2)
-  (comp-hom : composition-operation-binary-family-Set hom-set)
-  where
+### Unital composition operations on binary families of sets
 
-  is-associative-witness-associative-composition-operation-binary-family-Set :
-    ( {x y z w : A}
-      (h : type-Set (hom-set z w))
-      (g : type-Set (hom-set y z))
-      (f : type-Set (hom-set x y)) →
-      comp-hom (comp-hom h g) f ＝ comp-hom h (comp-hom g f)) →
-    is-associative-composition-operation-binary-family-Set hom-set comp-hom
-  pr1
-    ( is-associative-witness-associative-composition-operation-binary-family-Set
-        H h g f) =
-    H h g f
-  pr2
-    ( is-associative-witness-associative-composition-operation-binary-family-Set
-        H h g f) =
-    inv (H h g f)
-
-  is-associative-inv-witness-associative-composition-operation-binary-family-Set :
-    ( {x y z w : A}
-      (h : type-Set (hom-set z w))
-      (g : type-Set (hom-set y z))
-      (f : type-Set (hom-set x y)) →
-      comp-hom h (comp-hom g f) ＝ comp-hom (comp-hom h g) f) →
-    is-associative-composition-operation-binary-family-Set hom-set comp-hom
-  pr1
-    ( is-associative-inv-witness-associative-composition-operation-binary-family-Set
-        H h g f) =
-    inv (H h g f)
-  pr2
-    ( is-associative-inv-witness-associative-composition-operation-binary-family-Set
-        H h g f) =
-    H h g f
+A composition operation
+
+```text
+  _∘_ : hom y z → hom x y → hom x z
+```
+
+on a binary family of sets of morphisms is called
+{{#concept "unital" Disambiguation="composition operation on a binary family of sets" Agda=is-unital-composition-operation-binary-family-Set}}
+if there is a morphism `id_x : hom x x` for every element `x : A` such that
+
+```text
+  id_y ∘ f ＝ f   and f ∘ id_x = f.
 ```
 
-### Unital composition operations in binary families of sets
+As will be demonstrated momentarily, every composition operation on a binary
+family of sets is unital in [at most one](foundation.subterminal-types.md) way.
 
 ```agda
 module _
@@ -171,7 +185,7 @@ module _
 
 ## Properties
 
-### Being associative is a property of composition operations in binary families of sets
+### Being associative is a property of composition operations on binary families of sets
 
 ```agda
 module _
@@ -187,15 +201,12 @@ module _
       ( λ x y z w →
         is-prop-iterated-Π 3
           ( λ h g f →
-            is-prop-product
+            is-prop-equiv
+              ( equiv-eq-involutive-eq)
               ( is-set-type-Set
                 ( hom-set x w)
                 ( comp-hom (comp-hom h g) f)
-                ( comp-hom h (comp-hom g f)))
-              ( is-set-type-Set
-                ( hom-set x w)
-                ( comp-hom h (comp-hom g f))
-                ( comp-hom (comp-hom h g) f))))
+                ( comp-hom h (comp-hom g f)))))
 
   is-associative-prop-composition-operation-binary-family-Set : Prop (l1 ⊔ l2)
   pr1 is-associative-prop-composition-operation-binary-family-Set =
@@ -204,7 +215,7 @@ module _
     is-prop-is-associative-composition-operation-binary-family-Set
 ```
 
-### Being unital is a property of composition operations in binary families of sets
+### Being unital is a property of composition operations on binary families of sets
 
 **Proof:** Suppose `e e' : (x : A) → hom-set x x` are both right and left units
 with regard to composition. It is enough to show that `e ＝ e'` since the right
@@ -266,7 +277,8 @@ module _
 
 - [Set-magmoids](category-theory.set-magmoids.md) capture the structure of
   composition operations on binary families of sets.
-- [Precategories](category-theory.precategories.md) are associative and unital
-  composition operations on binary families of sets.
-- [Nonunital precategories](category-theory.nonunital-precategories.md) are
-  associative composition operations on binary families of sets.
+- [Precategories](category-theory.precategories.md) are the structure of an
+  associative and unital composition operation on a binary families of sets.
+- [Nonunital precategories](category-theory.nonunital-precategories.md) are the
+  structure of an associative composition operation on a binary families of
+  sets.
diff --git a/src/category-theory/conservative-functors-precategories.lagda.md b/src/category-theory/conservative-functors-precategories.lagda.md
index e88c6afb17..0116f42e94 100644
--- a/src/category-theory/conservative-functors-precategories.lagda.md
+++ b/src/category-theory/conservative-functors-precategories.lagda.md
@@ -29,7 +29,7 @@ isomorphism in `C`.
 
 ## Definitions
 
-### The predicate of being conservative
+### The predicate on functors of being conservative
 
 ```agda
 module _
diff --git a/src/category-theory/copresheaf-categories.lagda.md b/src/category-theory/copresheaf-categories.lagda.md
index 1a9b6d961e..584f90ada5 100644
--- a/src/category-theory/copresheaf-categories.lagda.md
+++ b/src/category-theory/copresheaf-categories.lagda.md
@@ -10,7 +10,6 @@ module category-theory.copresheaf-categories where
 open import category-theory.categories
 open import category-theory.category-of-functors-from-small-to-large-categories
 open import category-theory.functors-from-small-to-large-precategories
-open import category-theory.functors-precategories
 open import category-theory.large-categories
 open import category-theory.large-precategories
 open import category-theory.natural-transformations-functors-from-small-to-large-precategories
@@ -210,29 +209,6 @@ module _
       { Z}
       { W}
 
-  inv-associative-comp-hom-copresheaf-Precategory :
-    {l3 l4 l5 l6 : Level}
-    (X : copresheaf-Precategory l3)
-    (Y : copresheaf-Precategory l4)
-    (Z : copresheaf-Precategory l5)
-    (W : copresheaf-Precategory l6)
-    (h : hom-copresheaf-Precategory Z W)
-    (g : hom-copresheaf-Precategory Y Z)
-    (f : hom-copresheaf-Precategory X Y) →
-    comp-hom-copresheaf-Precategory X Z W
-      ( h)
-      ( comp-hom-copresheaf-Precategory X Y Z g f) ＝
-    comp-hom-copresheaf-Precategory X Y W
-      ( comp-hom-copresheaf-Precategory Y Z W h g)
-      ( f)
-  inv-associative-comp-hom-copresheaf-Precategory X Y Z W =
-    inv-associative-comp-hom-Large-Precategory
-      ( copresheaf-large-precategory-Precategory)
-      { X = X}
-      { Y}
-      { Z}
-      { W}
-
   left-unit-law-comp-hom-copresheaf-Precategory :
     {l3 l4 : Level}
     (X : copresheaf-Precategory l3)
diff --git a/src/category-theory/dependent-products-of-categories.lagda.md b/src/category-theory/dependent-products-of-categories.lagda.md
index 891865fc17..64742bf6c0 100644
--- a/src/category-theory/dependent-products-of-categories.lagda.md
+++ b/src/category-theory/dependent-products-of-categories.lagda.md
@@ -20,6 +20,7 @@ open import foundation.function-types
 open import foundation.functoriality-dependent-function-types
 open import foundation.identity-types
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -75,25 +76,25 @@ module _
     hom-Π-Category x z
   comp-hom-Π-Category = comp-hom-Category Π-Category
 
-  associative-comp-hom-Π-Category :
+  involutive-eq-associative-comp-hom-Π-Category :
     {x y z w : obj-Π-Category}
     (h : hom-Π-Category z w)
     (g : hom-Π-Category y z)
     (f : hom-Π-Category x y) →
-    comp-hom-Π-Category (comp-hom-Π-Category h g) f ＝
+    comp-hom-Π-Category (comp-hom-Π-Category h g) f ＝ⁱ
     comp-hom-Π-Category h (comp-hom-Π-Category g f)
-  associative-comp-hom-Π-Category =
-    associative-comp-hom-Category Π-Category
+  involutive-eq-associative-comp-hom-Π-Category =
+    involutive-eq-associative-comp-hom-Category Π-Category
 
-  inv-associative-comp-hom-Π-Category :
+  associative-comp-hom-Π-Category :
     {x y z w : obj-Π-Category}
     (h : hom-Π-Category z w)
     (g : hom-Π-Category y z)
     (f : hom-Π-Category x y) →
-    comp-hom-Π-Category h (comp-hom-Π-Category g f) ＝
-    comp-hom-Π-Category (comp-hom-Π-Category h g) f
-  inv-associative-comp-hom-Π-Category =
-    inv-associative-comp-hom-Category Π-Category
+    comp-hom-Π-Category (comp-hom-Π-Category h g) f ＝
+    comp-hom-Π-Category h (comp-hom-Π-Category g f)
+  associative-comp-hom-Π-Category =
+    associative-comp-hom-Category Π-Category
 
   associative-composition-operation-Π-Category :
     associative-composition-operation-binary-family-Set hom-set-Π-Category
diff --git a/src/category-theory/dependent-products-of-large-categories.lagda.md b/src/category-theory/dependent-products-of-large-categories.lagda.md
index 945769cffc..d674d884ab 100644
--- a/src/category-theory/dependent-products-of-large-categories.lagda.md
+++ b/src/category-theory/dependent-products-of-large-categories.lagda.md
@@ -17,6 +17,7 @@ open import foundation.function-extensionality
 open import foundation.functoriality-dependent-function-types
 open import foundation.identity-types
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -98,7 +99,7 @@ module _
   associative-comp-hom-Π-Large-Category =
     associative-comp-hom-Large-Category Π-Large-Category
 
-  inv-associative-comp-hom-Π-Large-Category :
+  involutive-eq-associative-comp-hom-Π-Large-Category :
     {l2 l3 l4 l5 : Level}
     {x : obj-Π-Large-Category l2}
     {y : obj-Π-Large-Category l3}
@@ -107,10 +108,10 @@ module _
     (h : hom-Π-Large-Category z w)
     (g : hom-Π-Large-Category y z)
     (f : hom-Π-Large-Category x y) →
-    comp-hom-Π-Large-Category h (comp-hom-Π-Large-Category g f) ＝
-    comp-hom-Π-Large-Category (comp-hom-Π-Large-Category h g) f
-  inv-associative-comp-hom-Π-Large-Category =
-    inv-associative-comp-hom-Large-Category Π-Large-Category
+    comp-hom-Π-Large-Category (comp-hom-Π-Large-Category h g) f ＝ⁱ
+    comp-hom-Π-Large-Category h (comp-hom-Π-Large-Category g f)
+  involutive-eq-associative-comp-hom-Π-Large-Category =
+    involutive-eq-associative-comp-hom-Large-Category Π-Large-Category
 
   id-hom-Π-Large-Category :
     {l2 : Level} {x : obj-Π-Large-Category l2} →
diff --git a/src/category-theory/dependent-products-of-large-precategories.lagda.md b/src/category-theory/dependent-products-of-large-precategories.lagda.md
index a66c1e1fe8..83d18e5945 100644
--- a/src/category-theory/dependent-products-of-large-precategories.lagda.md
+++ b/src/category-theory/dependent-products-of-large-precategories.lagda.md
@@ -16,6 +16,7 @@ open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
 open import foundation.universe-levels
 ```
@@ -62,7 +63,7 @@ module _
   comp-hom-Π-Large-Precategory f g i =
     comp-hom-Large-Precategory (C i) (f i) (g i)
 
-  associative-comp-hom-Π-Large-Precategory :
+  involutive-eq-associative-comp-hom-Π-Large-Precategory :
     {l2 l3 l4 l5 : Level}
     {x : obj-Π-Large-Precategory l2}
     {y : obj-Π-Large-Precategory l3}
@@ -71,13 +72,18 @@ module _
     (h : hom-Π-Large-Precategory z w)
     (g : hom-Π-Large-Precategory y z)
     (f : hom-Π-Large-Precategory x y) →
-    ( comp-hom-Π-Large-Precategory (comp-hom-Π-Large-Precategory h g) f) ＝
+    ( comp-hom-Π-Large-Precategory (comp-hom-Π-Large-Precategory h g) f) ＝ⁱ
     ( comp-hom-Π-Large-Precategory h (comp-hom-Π-Large-Precategory g f))
-  associative-comp-hom-Π-Large-Precategory h g f =
-    eq-htpy
-      ( λ i → associative-comp-hom-Large-Precategory (C i) (h i) (g i) (f i))
+  involutive-eq-associative-comp-hom-Π-Large-Precategory h g f =
+    involutive-eq-involutive-htpy
+      ( λ i →
+        involutive-eq-associative-comp-hom-Large-Precategory
+          ( C i)
+          ( h i)
+          ( g i)
+          ( f i))
 
-  inv-associative-comp-hom-Π-Large-Precategory :
+  associative-comp-hom-Π-Large-Precategory :
     {l2 l3 l4 l5 : Level}
     {x : obj-Π-Large-Precategory l2}
     {y : obj-Π-Large-Precategory l3}
@@ -86,12 +92,11 @@ module _
     (h : hom-Π-Large-Precategory z w)
     (g : hom-Π-Large-Precategory y z)
     (f : hom-Π-Large-Precategory x y) →
-    ( comp-hom-Π-Large-Precategory h (comp-hom-Π-Large-Precategory g f)) ＝
-    ( comp-hom-Π-Large-Precategory (comp-hom-Π-Large-Precategory h g) f)
-  inv-associative-comp-hom-Π-Large-Precategory h g f =
-    eq-htpy
-      ( λ i →
-        inv-associative-comp-hom-Large-Precategory (C i) (h i) (g i) (f i))
+    ( comp-hom-Π-Large-Precategory (comp-hom-Π-Large-Precategory h g) f) ＝
+    ( comp-hom-Π-Large-Precategory h (comp-hom-Π-Large-Precategory g f))
+  associative-comp-hom-Π-Large-Precategory h g f =
+    eq-involutive-eq
+      ( involutive-eq-associative-comp-hom-Π-Large-Precategory h g f)
 
   id-hom-Π-Large-Precategory :
     {l2 : Level} {x : obj-Π-Large-Precategory l2} → hom-Π-Large-Precategory x x
@@ -125,10 +130,8 @@ module _
     comp-hom-Π-Large-Precategory
   id-hom-Large-Precategory Π-Large-Precategory =
     id-hom-Π-Large-Precategory
-  associative-comp-hom-Large-Precategory Π-Large-Precategory =
-    associative-comp-hom-Π-Large-Precategory
-  inv-associative-comp-hom-Large-Precategory Π-Large-Precategory =
-    inv-associative-comp-hom-Π-Large-Precategory
+  involutive-eq-associative-comp-hom-Large-Precategory Π-Large-Precategory =
+    involutive-eq-associative-comp-hom-Π-Large-Precategory
   left-unit-law-comp-hom-Large-Precategory Π-Large-Precategory =
     left-unit-law-comp-hom-Π-Large-Precategory
   right-unit-law-comp-hom-Large-Precategory Π-Large-Precategory =
diff --git a/src/category-theory/dependent-products-of-precategories.lagda.md b/src/category-theory/dependent-products-of-precategories.lagda.md
index e947708637..6e7448dc99 100644
--- a/src/category-theory/dependent-products-of-precategories.lagda.md
+++ b/src/category-theory/dependent-products-of-precategories.lagda.md
@@ -17,6 +17,7 @@ open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
 open import foundation.universe-levels
 ```
@@ -54,33 +55,33 @@ module _
     hom-Π-Precategory x z
   comp-hom-Π-Precategory f g i = comp-hom-Precategory (C i) (f i) (g i)
 
-  associative-comp-hom-Π-Precategory :
+  involutive-eq-associative-comp-hom-Π-Precategory :
     {x y z w : obj-Π-Precategory}
     (h : hom-Π-Precategory z w)
     (g : hom-Π-Precategory y z)
     (f : hom-Π-Precategory x y) →
-    ( comp-hom-Π-Precategory (comp-hom-Π-Precategory h g) f) ＝
+    ( comp-hom-Π-Precategory (comp-hom-Π-Precategory h g) f) ＝ⁱ
     ( comp-hom-Π-Precategory h (comp-hom-Π-Precategory g f))
-  associative-comp-hom-Π-Precategory h g f =
-    eq-htpy (λ i → associative-comp-hom-Precategory (C i) (h i) (g i) (f i))
+  involutive-eq-associative-comp-hom-Π-Precategory h g f =
+    involutive-eq-involutive-htpy
+      ( λ i →
+        involutive-eq-associative-comp-hom-Precategory (C i) (h i) (g i) (f i))
 
-  inv-associative-comp-hom-Π-Precategory :
+  associative-comp-hom-Π-Precategory :
     {x y z w : obj-Π-Precategory}
     (h : hom-Π-Precategory z w)
     (g : hom-Π-Precategory y z)
     (f : hom-Π-Precategory x y) →
-    ( comp-hom-Π-Precategory h (comp-hom-Π-Precategory g f)) ＝
-    ( comp-hom-Π-Precategory (comp-hom-Π-Precategory h g) f)
-  inv-associative-comp-hom-Π-Precategory h g f =
-    eq-htpy (λ i → inv-associative-comp-hom-Precategory (C i) (h i) (g i) (f i))
+    ( comp-hom-Π-Precategory (comp-hom-Π-Precategory h g) f) ＝
+    ( comp-hom-Π-Precategory h (comp-hom-Π-Precategory g f))
+  associative-comp-hom-Π-Precategory h g f =
+    eq-involutive-eq (involutive-eq-associative-comp-hom-Π-Precategory h g f)
 
   associative-composition-operation-Π-Precategory :
     associative-composition-operation-binary-family-Set hom-set-Π-Precategory
   pr1 associative-composition-operation-Π-Precategory = comp-hom-Π-Precategory
-  pr1 (pr2 associative-composition-operation-Π-Precategory h g f) =
-    associative-comp-hom-Π-Precategory h g f
-  pr2 (pr2 associative-composition-operation-Π-Precategory h g f) =
-    inv-associative-comp-hom-Π-Precategory h g f
+  pr2 associative-composition-operation-Π-Precategory =
+    involutive-eq-associative-comp-hom-Π-Precategory
 
   id-hom-Π-Precategory : {x : obj-Π-Precategory} → hom-Π-Precategory x x
   id-hom-Π-Precategory i = id-hom-Precategory (C i)
diff --git a/src/category-theory/discrete-categories.lagda.md b/src/category-theory/discrete-categories.lagda.md
index 9769a562f9..f4a10cea33 100644
--- a/src/category-theory/discrete-categories.lagda.md
+++ b/src/category-theory/discrete-categories.lagda.md
@@ -12,6 +12,7 @@ open import category-theory.precategories
 open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -28,13 +29,13 @@ module _
   where
 
   discrete-precategory-Set : Precategory l l
-  pr1 discrete-precategory-Set = type-Set X
-  pr1 (pr2 discrete-precategory-Set) x y =
-    set-Prop (x ＝ y , is-set-type-Set X x y)
-  pr1 (pr1 (pr2 (pr2 discrete-precategory-Set))) = concat' _
-  pr1 (pr2 (pr1 (pr2 (pr2 discrete-precategory-Set))) refl refl refl) = refl
-  pr2 (pr2 (pr1 (pr2 (pr2 discrete-precategory-Set))) refl refl refl) = refl
-  pr1 (pr2 (pr2 (pr2 discrete-precategory-Set))) x = refl
-  pr1 (pr2 (pr2 (pr2 (pr2 discrete-precategory-Set)))) _ = right-unit
-  pr2 (pr2 (pr2 (pr2 (pr2 discrete-precategory-Set)))) _ = left-unit
+  discrete-precategory-Set =
+    make-Precategory
+      ( type-Set X)
+      ( λ x y → set-Prop (Id-Prop X x y))
+      ( λ p q → q ∙ p)
+      ( λ x → refl)
+      ( λ h g f → inv (assoc f g h))
+      ( λ _ → right-unit)
+      ( λ _ → left-unit)
 ```
diff --git a/src/category-theory/equivalences-of-precategories.lagda.md b/src/category-theory/equivalences-of-precategories.lagda.md
index 9ec17ab524..5478d066df 100644
--- a/src/category-theory/equivalences-of-precategories.lagda.md
+++ b/src/category-theory/equivalences-of-precategories.lagda.md
@@ -31,7 +31,7 @@ A [functor](category-theory.functors-precategories.md) `F : C → D` is an
 
 ## Definition
 
-### The predicate of being an equivalence of precategories
+### The predicate on functors of being an equivalence of precategories
 
 ```agda
 module _
diff --git a/src/category-theory/full-large-subcategories.lagda.md b/src/category-theory/full-large-subcategories.lagda.md
index d69460afb2..4b96c679d6 100644
--- a/src/category-theory/full-large-subcategories.lagda.md
+++ b/src/category-theory/full-large-subcategories.lagda.md
@@ -15,6 +15,7 @@ open import category-theory.large-precategories
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -130,7 +131,7 @@ module _
       ( large-precategory-Large-Category C)
       ( P)
 
-  inv-associative-comp-hom-Full-Large-Subcategory :
+  involutive-eq-associative-comp-hom-Full-Large-Subcategory :
     {l1 l2 l3 l4 : Level}
     (X : obj-Full-Large-Subcategory l1)
     (Y : obj-Full-Large-Subcategory l2)
@@ -139,14 +140,14 @@ module _
     (h : hom-Full-Large-Subcategory Z W)
     (g : hom-Full-Large-Subcategory Y Z)
     (f : hom-Full-Large-Subcategory X Y) →
-    comp-hom-Full-Large-Subcategory X Z W
-      ( h)
-      ( comp-hom-Full-Large-Subcategory X Y Z g f) ＝
     comp-hom-Full-Large-Subcategory X Y W
       ( comp-hom-Full-Large-Subcategory Y Z W h g)
-      ( f)
-  inv-associative-comp-hom-Full-Large-Subcategory =
-    inv-associative-comp-hom-Full-Large-Subprecategory
+      ( f) ＝ⁱ
+    comp-hom-Full-Large-Subcategory X Z W
+      ( h)
+      ( comp-hom-Full-Large-Subcategory X Y Z g f)
+  involutive-eq-associative-comp-hom-Full-Large-Subcategory =
+    involutive-eq-associative-comp-hom-Full-Large-Subprecategory
       ( large-precategory-Large-Category C)
       ( P)
 
diff --git a/src/category-theory/full-large-subprecategories.lagda.md b/src/category-theory/full-large-subprecategories.lagda.md
index 51ef6c9323..a3789dd83e 100644
--- a/src/category-theory/full-large-subprecategories.lagda.md
+++ b/src/category-theory/full-large-subprecategories.lagda.md
@@ -18,6 +18,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtype-identity-principle
 open import foundation.subtypes
 open import foundation.universe-levels
@@ -127,7 +128,7 @@ module _
   associative-comp-hom-Full-Large-Subprecategory X Y Z W =
     associative-comp-hom-Large-Precategory C
 
-  inv-associative-comp-hom-Full-Large-Subprecategory :
+  involutive-eq-associative-comp-hom-Full-Large-Subprecategory :
     {l1 l2 l3 l4 : Level}
     (X : obj-Full-Large-Subprecategory l1)
     (Y : obj-Full-Large-Subprecategory l2)
@@ -136,14 +137,14 @@ module _
     (h : hom-Full-Large-Subprecategory Z W)
     (g : hom-Full-Large-Subprecategory Y Z)
     (f : hom-Full-Large-Subprecategory X Y) →
-    comp-hom-Full-Large-Subprecategory X Z W
-      ( h)
-      ( comp-hom-Full-Large-Subprecategory X Y Z g f) ＝
     comp-hom-Full-Large-Subprecategory X Y W
       ( comp-hom-Full-Large-Subprecategory Y Z W h g)
-      ( f)
-  inv-associative-comp-hom-Full-Large-Subprecategory X Y Z W =
-    inv-associative-comp-hom-Large-Precategory C
+      ( f) ＝ⁱ
+    comp-hom-Full-Large-Subprecategory X Z W
+      ( h)
+      ( comp-hom-Full-Large-Subprecategory X Y Z g f)
+  involutive-eq-associative-comp-hom-Full-Large-Subprecategory X Y Z W =
+    involutive-eq-associative-comp-hom-Large-Precategory C
 
   left-unit-law-comp-hom-Full-Large-Subprecategory :
     {l1 l2 : Level}
@@ -184,14 +185,10 @@ module _
   id-hom-Large-Precategory
     large-precategory-Full-Large-Subprecategory {l1} {X} =
     id-hom-Full-Large-Subprecategory X
-  associative-comp-hom-Large-Precategory
-    large-precategory-Full-Large-Subprecategory
-    {l1} {l2} {l3} {l4} {X} {Y} {Z} {W} =
-    associative-comp-hom-Full-Large-Subprecategory X Y Z W
-  inv-associative-comp-hom-Large-Precategory
+  involutive-eq-associative-comp-hom-Large-Precategory
     large-precategory-Full-Large-Subprecategory
     {l1} {l2} {l3} {l4} {X} {Y} {Z} {W} =
-    inv-associative-comp-hom-Full-Large-Subprecategory X Y Z W
+    involutive-eq-associative-comp-hom-Full-Large-Subprecategory X Y Z W
   left-unit-law-comp-hom-Large-Precategory
     large-precategory-Full-Large-Subprecategory {l1} {l2} {X} {Y} =
     left-unit-law-comp-hom-Full-Large-Subprecategory X Y
diff --git a/src/category-theory/full-subcategories.lagda.md b/src/category-theory/full-subcategories.lagda.md
index 6bd710e242..a1921d9182 100644
--- a/src/category-theory/full-subcategories.lagda.md
+++ b/src/category-theory/full-subcategories.lagda.md
@@ -21,6 +21,7 @@ open import foundation.embeddings
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
 open import foundation.universe-levels
 ```
@@ -129,19 +130,19 @@ module _
     associative-comp-hom-Full-Subprecategory
       ( precategory-Category C) P {x} {y} {z} {w}
 
-  inv-associative-comp-hom-Full-Subcategory :
+  involutive-eq-associative-comp-hom-Full-Subcategory :
     {x y z w : obj-Full-Subcategory C P}
     (h : hom-Full-Subcategory z w)
     (g : hom-Full-Subcategory y z)
     (f : hom-Full-Subcategory x y) →
-    comp-hom-Full-Subcategory {x} {z} {w}
-      ( h)
-      ( comp-hom-Full-Subcategory {x} {y} {z} g f) ＝
     comp-hom-Full-Subcategory {x} {y} {w}
       ( comp-hom-Full-Subcategory {y} {z} {w} h g)
-      ( f)
-  inv-associative-comp-hom-Full-Subcategory {x} {y} {z} {w} =
-    inv-associative-comp-hom-Full-Subprecategory
+      ( f) ＝ⁱ
+    comp-hom-Full-Subcategory {x} {z} {w}
+      ( h)
+      ( comp-hom-Full-Subcategory {x} {y} {z} g f)
+  involutive-eq-associative-comp-hom-Full-Subcategory {x} {y} {z} {w} =
+    involutive-eq-associative-comp-hom-Full-Subprecategory
       ( precategory-Category C) P {x} {y} {z} {w}
 
   left-unit-law-comp-hom-Full-Subcategory :
diff --git a/src/category-theory/full-subprecategories.lagda.md b/src/category-theory/full-subprecategories.lagda.md
index f959b45d70..7d2faeebad 100644
--- a/src/category-theory/full-subprecategories.lagda.md
+++ b/src/category-theory/full-subprecategories.lagda.md
@@ -25,6 +25,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtype-identity-principle
 open import foundation.subtypes
 open import foundation.universe-levels
@@ -118,33 +119,33 @@ module _
     hom-Full-Subprecategory x z
   comp-hom-Full-Subprecategory = comp-hom-Precategory C
 
-  associative-comp-hom-Full-Subprecategory :
+  involutive-eq-associative-comp-hom-Full-Subprecategory :
     {x y z w : obj-Full-Subprecategory C P}
     (h : hom-Full-Subprecategory z w)
     (g : hom-Full-Subprecategory y z)
     (f : hom-Full-Subprecategory x y) →
     comp-hom-Full-Subprecategory {x} {y} {w}
       ( comp-hom-Full-Subprecategory {y} {z} {w} h g)
-      ( f) ＝
+      ( f) ＝ⁱ
     comp-hom-Full-Subprecategory {x} {z} {w}
       ( h)
       ( comp-hom-Full-Subprecategory {x} {y} {z} g f)
-  associative-comp-hom-Full-Subprecategory =
-    associative-comp-hom-Precategory C
+  involutive-eq-associative-comp-hom-Full-Subprecategory =
+    involutive-eq-associative-comp-hom-Precategory C
 
-  inv-associative-comp-hom-Full-Subprecategory :
+  associative-comp-hom-Full-Subprecategory :
     {x y z w : obj-Full-Subprecategory C P}
     (h : hom-Full-Subprecategory z w)
     (g : hom-Full-Subprecategory y z)
     (f : hom-Full-Subprecategory x y) →
-    comp-hom-Full-Subprecategory {x} {z} {w}
-      ( h)
-      ( comp-hom-Full-Subprecategory {x} {y} {z} g f) ＝
     comp-hom-Full-Subprecategory {x} {y} {w}
       ( comp-hom-Full-Subprecategory {y} {z} {w} h g)
-      ( f)
-  inv-associative-comp-hom-Full-Subprecategory =
-    inv-associative-comp-hom-Precategory C
+      ( f) ＝
+    comp-hom-Full-Subprecategory {x} {z} {w}
+      ( h)
+      ( comp-hom-Full-Subprecategory {x} {y} {z} g f)
+  associative-comp-hom-Full-Subprecategory =
+    associative-comp-hom-Precategory C
 
   left-unit-law-comp-hom-Full-Subprecategory :
     {x y : obj-Full-Subprecategory C P}
@@ -171,16 +172,10 @@ module _
       hom-set-Full-Subprecategory
   pr1 associative-composition-operation-Full-Subprecategory {x} {y} {z} =
     comp-hom-Full-Subprecategory {x} {y} {z}
-  pr1
-    ( pr2
-        associative-composition-operation-Full-Subprecategory
-          { x} {y} {z} {w} h g f) =
-    associative-comp-hom-Full-Subprecategory {x} {y} {z} {w} h g f
   pr2
-    ( pr2
-        associative-composition-operation-Full-Subprecategory
-          { x} {y} {z} {w} h g f) =
-    inv-associative-comp-hom-Full-Subprecategory {x} {y} {z} {w} h g f
+    associative-composition-operation-Full-Subprecategory
+    { x} {y} {z} {w} h g f =
+    involutive-eq-associative-comp-hom-Full-Subprecategory {x} {y} {z} {w} h g f
 
   is-unital-composition-operation-Full-Subprecategory :
     is-unital-composition-operation-binary-family-Set
diff --git a/src/category-theory/function-categories.lagda.md b/src/category-theory/function-categories.lagda.md
index 0b69fe58ce..2e86a66718 100644
--- a/src/category-theory/function-categories.lagda.md
+++ b/src/category-theory/function-categories.lagda.md
@@ -16,6 +16,7 @@ open import category-theory.precategories
 open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -73,15 +74,15 @@ module _
   associative-comp-hom-function-Category =
     associative-comp-hom-Category function-Category
 
-  inv-associative-comp-hom-function-Category :
+  involutive-eq-associative-comp-hom-function-Category :
     {x y z w : obj-function-Category}
     (h : hom-function-Category z w)
     (g : hom-function-Category y z)
     (f : hom-function-Category x y) →
-    comp-hom-function-Category h (comp-hom-function-Category g f) ＝
-    comp-hom-function-Category (comp-hom-function-Category h g) f
-  inv-associative-comp-hom-function-Category =
-    inv-associative-comp-hom-Category function-Category
+    comp-hom-function-Category (comp-hom-function-Category h g) f ＝ⁱ
+    comp-hom-function-Category h (comp-hom-function-Category g f)
+  involutive-eq-associative-comp-hom-function-Category =
+    involutive-eq-associative-comp-hom-Category function-Category
 
   associative-composition-operation-function-Category :
     associative-composition-operation-binary-family-Set
diff --git a/src/category-theory/function-precategories.lagda.md b/src/category-theory/function-precategories.lagda.md
index ddbcfb33d6..64506a827d 100644
--- a/src/category-theory/function-precategories.lagda.md
+++ b/src/category-theory/function-precategories.lagda.md
@@ -15,6 +15,7 @@ open import category-theory.precategories
 open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -65,15 +66,15 @@ module _
   associative-comp-hom-function-Precategory =
     associative-comp-hom-Precategory function-Precategory
 
-  inv-associative-comp-hom-function-Precategory :
+  involutive-eq-associative-comp-hom-function-Precategory :
     {x y z w : obj-function-Precategory}
     (h : hom-function-Precategory z w)
     (g : hom-function-Precategory y z)
     (f : hom-function-Precategory x y) →
-    comp-hom-function-Precategory h (comp-hom-function-Precategory g f) ＝
-    comp-hom-function-Precategory (comp-hom-function-Precategory h g) f
-  inv-associative-comp-hom-function-Precategory =
-    inv-associative-comp-hom-Precategory function-Precategory
+    comp-hom-function-Precategory (comp-hom-function-Precategory h g) f ＝ⁱ
+    comp-hom-function-Precategory h (comp-hom-function-Precategory g f)
+  involutive-eq-associative-comp-hom-function-Precategory =
+    involutive-eq-associative-comp-hom-Precategory function-Precategory
 
   associative-composition-operation-function-Precategory :
     associative-composition-operation-binary-family-Set
diff --git a/src/category-theory/gaunt-categories.lagda.md b/src/category-theory/gaunt-categories.lagda.md
index 8e495012bb..8235795ebe 100644
--- a/src/category-theory/gaunt-categories.lagda.md
+++ b/src/category-theory/gaunt-categories.lagda.md
@@ -23,6 +23,7 @@ open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -166,15 +167,15 @@ module _
   associative-comp-hom-Gaunt-Category =
     associative-comp-hom-Category category-Gaunt-Category
 
-  inv-associative-comp-hom-Gaunt-Category :
+  involutive-eq-associative-comp-hom-Gaunt-Category :
     {x y z w : obj-Gaunt-Category}
     (h : hom-Gaunt-Category z w)
     (g : hom-Gaunt-Category y z)
     (f : hom-Gaunt-Category x y) →
-    comp-hom-Gaunt-Category h (comp-hom-Gaunt-Category g f) ＝
-    comp-hom-Gaunt-Category (comp-hom-Gaunt-Category h g) f
-  inv-associative-comp-hom-Gaunt-Category =
-    inv-associative-comp-hom-Category category-Gaunt-Category
+    comp-hom-Gaunt-Category (comp-hom-Gaunt-Category h g) f ＝ⁱ
+    comp-hom-Gaunt-Category h (comp-hom-Gaunt-Category g f)
+  involutive-eq-associative-comp-hom-Gaunt-Category =
+    involutive-eq-associative-comp-hom-Category category-Gaunt-Category
 
   associative-composition-operation-Gaunt-Category :
     associative-composition-operation-binary-family-Set
diff --git a/src/category-theory/groupoids.lagda.md b/src/category-theory/groupoids.lagda.md
index 7028c7da72..117845dff7 100644
--- a/src/category-theory/groupoids.lagda.md
+++ b/src/category-theory/groupoids.lagda.md
@@ -25,6 +25,7 @@ open import foundation.identity-types
 open import foundation.iterated-dependent-pair-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 ```
@@ -88,13 +89,13 @@ module _
   associative-comp-hom-Groupoid =
     associative-comp-hom-Category category-Groupoid
 
-  inv-associative-comp-hom-Groupoid :
+  involutive-eq-associative-comp-hom-Groupoid :
     {x y z w : obj-Groupoid}
     (h : hom-Groupoid z w) (g : hom-Groupoid y z) (f : hom-Groupoid x y) →
-    comp-hom-Groupoid h (comp-hom-Groupoid g f) ＝
-    comp-hom-Groupoid (comp-hom-Groupoid h g) f
-  inv-associative-comp-hom-Groupoid =
-    inv-associative-comp-hom-Category category-Groupoid
+    comp-hom-Groupoid (comp-hom-Groupoid h g) f ＝ⁱ
+    comp-hom-Groupoid h (comp-hom-Groupoid g f)
+  involutive-eq-associative-comp-hom-Groupoid =
+    involutive-eq-associative-comp-hom-Category category-Groupoid
 
   left-unit-law-comp-hom-Groupoid :
     {x y : obj-Groupoid} (f : hom-Groupoid x y) →
@@ -130,16 +131,15 @@ module _
   obj-groupoid-1-Type = type-1-Type X
 
   precategory-Groupoid-1-Type : Precategory l l
-  pr1 precategory-Groupoid-1-Type = obj-groupoid-1-Type
-  pr1 (pr2 precategory-Groupoid-1-Type) = Id-Set X
-  pr1 (pr1 (pr2 (pr2 precategory-Groupoid-1-Type))) q p = p ∙ q
-  pr1 (pr2 (pr1 (pr2 (pr2 precategory-Groupoid-1-Type))) r q p) =
-    inv (assoc p q r)
-  pr2 (pr2 (pr1 (pr2 (pr2 precategory-Groupoid-1-Type))) r q p) =
-    assoc p q r
-  pr1 (pr2 (pr2 (pr2 precategory-Groupoid-1-Type))) x = refl
-  pr1 (pr2 (pr2 (pr2 (pr2 precategory-Groupoid-1-Type)))) p = right-unit
-  pr2 (pr2 (pr2 (pr2 (pr2 precategory-Groupoid-1-Type)))) p = left-unit
+  precategory-Groupoid-1-Type =
+    make-Precategory
+      ( obj-groupoid-1-Type)
+      ( Id-Set X)
+      ( λ q p → p ∙ q)
+      ( λ x → refl)
+      ( λ r q p → inv (assoc p q r))
+      ( λ p → right-unit)
+      ( λ p → left-unit)
 
   is-category-groupoid-1-Type :
     is-category-Precategory precategory-Groupoid-1-Type
diff --git a/src/category-theory/indiscrete-precategories.lagda.md b/src/category-theory/indiscrete-precategories.lagda.md
index 0ac8c5fe4a..68479adbd5 100644
--- a/src/category-theory/indiscrete-precategories.lagda.md
+++ b/src/category-theory/indiscrete-precategories.lagda.md
@@ -7,29 +7,22 @@ module category-theory.indiscrete-precategories where
 <details><summary>Imports</summary>
 
 ```agda
-open import category-theory.composition-operations-on-binary-families-of-sets
-open import category-theory.fully-faithful-functors-precategories
 open import category-theory.isomorphisms-in-precategories
 open import category-theory.precategories
 open import category-theory.pregroupoids
 open import category-theory.preunivalent-categories
 open import category-theory.strict-categories
 open import category-theory.subterminal-precategories
-open import category-theory.terminal-category
 
-open import foundation.action-on-identifications-functions
 open import foundation.contractible-types
 open import foundation.dependent-pair-types
-open import foundation.embeddings
 open import foundation.equivalences
 open import foundation.function-types
-open import foundation.fundamental-theorem-of-identity-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.iterated-dependent-product-types
 open import foundation.propositions
 open import foundation.sets
-open import foundation.subtype-identity-principle
 open import foundation.unit-type
 open import foundation.universe-levels
 ```
@@ -98,35 +91,6 @@ module _
       ( comp-hom-indiscrete-Precategory {x} {y} {z} g f)
   associative-comp-hom-indiscrete-Precategory h g f = refl
 
-  inv-associative-comp-hom-indiscrete-Precategory :
-    {x y z w : obj-indiscrete-Precategory X} →
-    (h : hom-indiscrete-Precategory X z w)
-    (g : hom-indiscrete-Precategory X y z)
-    (f : hom-indiscrete-Precategory X x y) →
-    comp-hom-indiscrete-Precategory {x} {z} {w}
-      ( h)
-      ( comp-hom-indiscrete-Precategory {x} {y} {z} g f) ＝
-    comp-hom-indiscrete-Precategory {x} {y} {w}
-      ( comp-hom-indiscrete-Precategory {y} {z} {w} h g)
-      ( f)
-  inv-associative-comp-hom-indiscrete-Precategory h g f = refl
-
-  associative-composition-operation-indiscrete-Precategory :
-    associative-composition-operation-binary-family-Set
-      ( hom-set-indiscrete-Precategory X)
-  pr1 associative-composition-operation-indiscrete-Precategory {x} {y} {z} =
-    comp-hom-indiscrete-Precategory {x} {y} {z}
-  pr1
-    ( pr2
-        associative-composition-operation-indiscrete-Precategory
-          { x} {y} {z} {w} h g f) =
-    associative-comp-hom-indiscrete-Precategory {x} {y} {z} {w} h g f
-  pr2
-    ( pr2
-        associative-composition-operation-indiscrete-Precategory
-          { x} {y} {z} {w} h g f) =
-    inv-associative-comp-hom-indiscrete-Precategory {x} {y} {z} {w} h g f
-
   id-hom-indiscrete-Precategory :
     {x : obj-indiscrete-Precategory X} → hom-indiscrete-Precategory X x x
   id-hom-indiscrete-Precategory = star
@@ -149,24 +113,17 @@ module _
     f
   right-unit-law-comp-hom-indiscrete-Precategory f = refl
 
-  is-unital-composition-operation-indiscrete-Precategory :
-    is-unital-composition-operation-binary-family-Set
+  indiscrete-Precategory : Precategory l lzero
+  indiscrete-Precategory =
+    make-Precategory
+      ( obj-indiscrete-Precategory X)
       ( hom-set-indiscrete-Precategory X)
       ( λ {x} {y} {z} → comp-hom-indiscrete-Precategory {x} {y} {z})
-  pr1 is-unital-composition-operation-indiscrete-Precategory x =
-    id-hom-indiscrete-Precategory {x}
-  pr1 (pr2 is-unital-composition-operation-indiscrete-Precategory) {x} {y} =
-    left-unit-law-comp-hom-indiscrete-Precategory {x} {y}
-  pr2 (pr2 is-unital-composition-operation-indiscrete-Precategory) {x} {y} =
-    right-unit-law-comp-hom-indiscrete-Precategory {x} {y}
-
-  indiscrete-Precategory : Precategory l lzero
-  pr1 indiscrete-Precategory = obj-indiscrete-Precategory X
-  pr1 (pr2 indiscrete-Precategory) = hom-set-indiscrete-Precategory X
-  pr1 (pr2 (pr2 indiscrete-Precategory)) =
-    associative-composition-operation-indiscrete-Precategory
-  pr2 (pr2 (pr2 indiscrete-Precategory)) =
-    is-unital-composition-operation-indiscrete-Precategory
+      ( λ x → id-hom-indiscrete-Precategory {x})
+      ( λ {x} {y} {z} {w} →
+        associative-comp-hom-indiscrete-Precategory {x} {y} {z} {w})
+      ( λ {x} {y} → left-unit-law-comp-hom-indiscrete-Precategory {x} {y})
+      ( λ {x} {y} → right-unit-law-comp-hom-indiscrete-Precategory {x} {y})
 ```
 
 #### The pregroupoid structure of the indiscrete precategory associated to a type
diff --git a/src/category-theory/initial-category.lagda.md b/src/category-theory/initial-category.lagda.md
index b0976ba941..14a42fb3f9 100644
--- a/src/category-theory/initial-category.lagda.md
+++ b/src/category-theory/initial-category.lagda.md
@@ -51,25 +51,42 @@ hom-initial-Category x y = type-Set (hom-set-initial-Category x y)
 ### The underlying precategory of the initial category
 
 ```agda
-comp-hom-initial-Category =
-  comp-hom-indiscrete-Precategory empty
-
-associative-comp-hom-initial-Category =
-  associative-comp-hom-indiscrete-Precategory empty
-
-associative-composition-operation-initial-Category =
-  associative-composition-operation-indiscrete-Precategory empty
-
-id-hom-initial-Category = id-hom-indiscrete-Precategory empty
-
-left-unit-law-comp-hom-initial-Category =
-  left-unit-law-comp-hom-indiscrete-Precategory empty
-
-right-unit-law-comp-hom-initial-Category =
-  right-unit-law-comp-hom-indiscrete-Precategory empty
-
-is-unital-composition-operation-initial-Category =
-  is-unital-composition-operation-indiscrete-Precategory empty
+comp-hom-initial-Category :
+  {x y z : obj-initial-Category} →
+  hom-initial-Category y z → hom-initial-Category x y → hom-initial-Category x z
+comp-hom-initial-Category {x} {y} {z} =
+  comp-hom-indiscrete-Precategory empty {x} {y} {z}
+
+associative-comp-hom-initial-Category :
+  {x y z w : obj-initial-Category}
+  (h : hom-initial-Category z w)
+  (g : hom-initial-Category y z)
+  (f : hom-initial-Category x y) →
+  comp-hom-initial-Category {x} {y} {w}
+    ( comp-hom-initial-Category {y} {z} {w} h g)
+    ( f) ＝
+  comp-hom-initial-Category {x} {z} {w}
+    ( h)
+    ( comp-hom-initial-Category {x} {y} {z} g f)
+associative-comp-hom-initial-Category {x} {y} {z} {w} =
+  associative-comp-hom-indiscrete-Precategory empty {x} {y} {z} {w}
+
+id-hom-initial-Category : {x : obj-initial-Category} → hom-initial-Category x x
+id-hom-initial-Category {x} = id-hom-indiscrete-Precategory empty {x}
+
+left-unit-law-comp-hom-initial-Category :
+  {x y : obj-initial-Category}
+  (f : hom-initial-Category x y) →
+  comp-hom-initial-Category {x} {y} {y} (id-hom-initial-Category {y}) f ＝ f
+left-unit-law-comp-hom-initial-Category {x} {y} =
+  left-unit-law-comp-hom-indiscrete-Precategory empty {x} {y}
+
+right-unit-law-comp-hom-initial-Category :
+  {x y : obj-initial-Category}
+  (f : hom-initial-Category x y) →
+  comp-hom-initial-Category {x} {x} {y} f (id-hom-initial-Category {x}) ＝ f
+right-unit-law-comp-hom-initial-Category {x} {y} =
+  right-unit-law-comp-hom-indiscrete-Precategory empty {x} {y}
 
 initial-Precategory : Precategory lzero lzero
 initial-Precategory = indiscrete-Precategory empty
diff --git a/src/category-theory/large-categories.lagda.md b/src/category-theory/large-categories.lagda.md
index c391818c5e..b798eee311 100644
--- a/src/category-theory/large-categories.lagda.md
+++ b/src/category-theory/large-categories.lagda.md
@@ -18,6 +18,7 @@ open import foundation.equivalences
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -117,7 +118,7 @@ module _
   id-hom-Large-Category =
     id-hom-Large-Precategory (large-precategory-Large-Category C)
 
-  associative-comp-hom-Large-Category :
+  involutive-eq-associative-comp-hom-Large-Category :
     {l1 l2 l3 l4 : Level}
     {X : obj-Large-Category l1}
     {Y : obj-Large-Category l2}
@@ -126,12 +127,13 @@ module _
     (h : hom-Large-Category Z W)
     (g : hom-Large-Category Y Z)
     (f : hom-Large-Category X Y) →
-    ( comp-hom-Large-Category (comp-hom-Large-Category h g) f) ＝
+    ( comp-hom-Large-Category (comp-hom-Large-Category h g) f) ＝ⁱ
     ( comp-hom-Large-Category h (comp-hom-Large-Category g f))
-  associative-comp-hom-Large-Category =
-    associative-comp-hom-Large-Precategory (large-precategory-Large-Category C)
+  involutive-eq-associative-comp-hom-Large-Category =
+    involutive-eq-associative-comp-hom-Large-Precategory
+      ( large-precategory-Large-Category C)
 
-  inv-associative-comp-hom-Large-Category :
+  associative-comp-hom-Large-Category :
     {l1 l2 l3 l4 : Level}
     {X : obj-Large-Category l1}
     {Y : obj-Large-Category l2}
@@ -140,11 +142,10 @@ module _
     (h : hom-Large-Category Z W)
     (g : hom-Large-Category Y Z)
     (f : hom-Large-Category X Y) →
-    ( comp-hom-Large-Category h (comp-hom-Large-Category g f)) ＝
-    ( comp-hom-Large-Category (comp-hom-Large-Category h g) f)
-  inv-associative-comp-hom-Large-Category =
-    inv-associative-comp-hom-Large-Precategory
-      ( large-precategory-Large-Category C)
+    ( comp-hom-Large-Category (comp-hom-Large-Category h g) f) ＝
+    ( comp-hom-Large-Category h (comp-hom-Large-Category g f))
+  associative-comp-hom-Large-Category =
+    associative-comp-hom-Large-Precategory (large-precategory-Large-Category C)
 
   left-unit-law-comp-hom-Large-Category :
     {l1 l2 : Level}
diff --git a/src/category-theory/large-function-categories.lagda.md b/src/category-theory/large-function-categories.lagda.md
index 9d133cf142..dedadafc7f 100644
--- a/src/category-theory/large-function-categories.lagda.md
+++ b/src/category-theory/large-function-categories.lagda.md
@@ -14,6 +14,7 @@ open import category-theory.large-categories
 open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -22,8 +23,9 @@ open import foundation.universe-levels
 ## Idea
 
 Given a type `I` and a [large category](category-theory.large-categories.md)
-`C`, the **large function category** `Cᴵ` consists of `I`-indexed families of
-objects of `C` and `I`-indexed familis of morphisms between them.
+`C`, the {{#concept "large function category" Agda=Large-Function-Category}}
+`Cᴵ` consists of `I`-indexed families of objects of `C` and `I`-indexed families
+of morphisms between them.
 
 ## Definition
 
@@ -83,7 +85,7 @@ module _
   associative-comp-hom-Large-Function-Category =
     associative-comp-hom-Π-Large-Category I (λ _ → C)
 
-  inv-associative-comp-hom-Large-Function-Category :
+  involutive-eq-associative-comp-hom-Large-Function-Category :
     {l2 l3 l4 l5 : Level}
     {x : obj-Large-Function-Category l2}
     {y : obj-Large-Function-Category l3}
@@ -92,10 +94,14 @@ module _
     (h : hom-Large-Function-Category z w)
     (g : hom-Large-Function-Category y z)
     (f : hom-Large-Function-Category x y) →
-    comp-hom-Large-Function-Category h (comp-hom-Large-Function-Category g f) ＝
-    comp-hom-Large-Function-Category (comp-hom-Large-Function-Category h g) f
-  inv-associative-comp-hom-Large-Function-Category =
-    inv-associative-comp-hom-Π-Large-Category I (λ _ → C)
+    comp-hom-Large-Function-Category
+      ( comp-hom-Large-Function-Category h g)
+      ( f) ＝ⁱ
+    comp-hom-Large-Function-Category
+      ( h)
+      ( comp-hom-Large-Function-Category g f)
+  involutive-eq-associative-comp-hom-Large-Function-Category =
+    involutive-eq-associative-comp-hom-Π-Large-Category I (λ _ → C)
 
   id-hom-Large-Function-Category :
     {l2 : Level} {x : obj-Large-Function-Category l2} →
diff --git a/src/category-theory/large-function-precategories.lagda.md b/src/category-theory/large-function-precategories.lagda.md
index 1314c5319e..e7b592b608 100644
--- a/src/category-theory/large-function-precategories.lagda.md
+++ b/src/category-theory/large-function-precategories.lagda.md
@@ -14,6 +14,7 @@ open import category-theory.large-precategories
 open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -24,7 +25,7 @@ open import foundation.universe-levels
 Given a type `I` and a
 [large precategory](category-theory.large-precategories.md) `C`, the **large
 function pre-category** `Cᴵ` consists of `I`-indexed families of objects of `C`
-and `I`-indexed familis of morphisms between them.
+and `I`-indexed families of morphisms between them.
 
 ## Definition
 
@@ -88,7 +89,7 @@ module _
   associative-comp-hom-Large-Function-Precategory =
     associative-comp-hom-Π-Large-Precategory I (λ _ → C)
 
-  inv-associative-comp-hom-Large-Function-Precategory :
+  involutive-eq-associative-comp-hom-Large-Function-Precategory :
     {l2 l3 l4 l5 : Level}
     {x : obj-Large-Function-Precategory l2}
     {y : obj-Large-Function-Precategory l3}
@@ -97,14 +98,14 @@ module _
     (h : hom-Large-Function-Precategory z w)
     (g : hom-Large-Function-Precategory y z)
     (f : hom-Large-Function-Precategory x y) →
-    comp-hom-Large-Function-Precategory
-      ( h)
-      ( comp-hom-Large-Function-Precategory g f) ＝
     comp-hom-Large-Function-Precategory
       ( comp-hom-Large-Function-Precategory h g)
-      ( f)
-  inv-associative-comp-hom-Large-Function-Precategory =
-    inv-associative-comp-hom-Π-Large-Precategory I (λ _ → C)
+      ( f) ＝ⁱ
+    comp-hom-Large-Function-Precategory
+      ( h)
+      ( comp-hom-Large-Function-Precategory g f)
+  involutive-eq-associative-comp-hom-Large-Function-Precategory =
+    involutive-eq-associative-comp-hom-Π-Large-Precategory I (λ _ → C)
 
   id-hom-Large-Function-Precategory :
     {l2 : Level} {x : obj-Large-Function-Precategory l2} →
diff --git a/src/category-theory/large-precategories.lagda.md b/src/category-theory/large-precategories.lagda.md
index 837dbd4f09..da7bed345a 100644
--- a/src/category-theory/large-precategories.lagda.md
+++ b/src/category-theory/large-precategories.lagda.md
@@ -15,6 +15,7 @@ open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -33,7 +34,6 @@ be done with Σ-types, we must use a record type.)
 ```agda
 record
   Large-Precategory (α : Level → Level) (β : Level → Level → Level) : UUω where
-  constructor make-Large-Precategory
   field
     obj-Large-Precategory :
       (l : Level) → UU (α l)
@@ -44,59 +44,115 @@ record
       obj-Large-Precategory l2 →
       Set (β l1 l2)
 
+  hom-Large-Precategory :
+    {l1 l2 : Level}
+    (X : obj-Large-Precategory l1)
+    (Y : obj-Large-Precategory l2) →
+    UU (β l1 l2)
+  hom-Large-Precategory X Y = type-Set (hom-set-Large-Precategory X Y)
+
+  is-set-hom-Large-Precategory :
+    {l1 l2 : Level}
+    (X : obj-Large-Precategory l1)
+    (Y : obj-Large-Precategory l2) →
+    is-set (hom-Large-Precategory X Y)
+  is-set-hom-Large-Precategory X Y =
+    is-set-type-Set (hom-set-Large-Precategory X Y)
+
+  field
     comp-hom-Large-Precategory :
       {l1 l2 l3 : Level}
       {X : obj-Large-Precategory l1}
       {Y : obj-Large-Precategory l2}
       {Z : obj-Large-Precategory l3} →
-      type-Set (hom-set-Large-Precategory Y Z) →
-      type-Set (hom-set-Large-Precategory X Y) →
-      type-Set (hom-set-Large-Precategory X Z)
+      hom-Large-Precategory Y Z →
+      hom-Large-Precategory X Y →
+      hom-Large-Precategory X Z
 
     id-hom-Large-Precategory :
       {l1 : Level}
       {X : obj-Large-Precategory l1} →
-      type-Set (hom-set-Large-Precategory X X)
+      hom-Large-Precategory X X
 
-    associative-comp-hom-Large-Precategory :
+    involutive-eq-associative-comp-hom-Large-Precategory :
       {l1 l2 l3 l4 : Level}
       {X : obj-Large-Precategory l1}
       {Y : obj-Large-Precategory l2}
       {Z : obj-Large-Precategory l3}
       {W : obj-Large-Precategory l4} →
-      (h : type-Set (hom-set-Large-Precategory Z W))
-      (g : type-Set (hom-set-Large-Precategory Y Z))
-      (f : type-Set (hom-set-Large-Precategory X Y)) →
-      ( comp-hom-Large-Precategory (comp-hom-Large-Precategory h g) f) ＝
+      (h : hom-Large-Precategory Z W)
+      (g : hom-Large-Precategory Y Z)
+      (f : hom-Large-Precategory X Y) →
+      ( comp-hom-Large-Precategory (comp-hom-Large-Precategory h g) f) ＝ⁱ
       ( comp-hom-Large-Precategory h (comp-hom-Large-Precategory g f))
 
-    inv-associative-comp-hom-Large-Precategory :
-      {l1 l2 l3 l4 : Level}
-      {X : obj-Large-Precategory l1}
-      {Y : obj-Large-Precategory l2}
-      {Z : obj-Large-Precategory l3}
-      {W : obj-Large-Precategory l4} →
-      (h : type-Set (hom-set-Large-Precategory Z W))
-      (g : type-Set (hom-set-Large-Precategory Y Z))
-      (f : type-Set (hom-set-Large-Precategory X Y)) →
-      ( comp-hom-Large-Precategory h (comp-hom-Large-Precategory g f)) ＝
-      ( comp-hom-Large-Precategory (comp-hom-Large-Precategory h g) f)
-
     left-unit-law-comp-hom-Large-Precategory :
       {l1 l2 : Level}
       {X : obj-Large-Precategory l1}
       {Y : obj-Large-Precategory l2}
-      (f : type-Set (hom-set-Large-Precategory X Y)) →
+      (f : hom-Large-Precategory X Y) →
       ( comp-hom-Large-Precategory id-hom-Large-Precategory f) ＝ f
 
     right-unit-law-comp-hom-Large-Precategory :
       {l1 l2 : Level}
       {X : obj-Large-Precategory l1}
       {Y : obj-Large-Precategory l2}
-      (f : type-Set (hom-set-Large-Precategory X Y)) →
+      (f : hom-Large-Precategory X Y) →
       ( comp-hom-Large-Precategory f id-hom-Large-Precategory) ＝ f
 
+  associative-comp-hom-Large-Precategory :
+      {l1 l2 l3 l4 : Level}
+      {X : obj-Large-Precategory l1}
+      {Y : obj-Large-Precategory l2}
+      {Z : obj-Large-Precategory l3}
+      {W : obj-Large-Precategory l4} →
+      (h : hom-Large-Precategory Z W)
+      (g : hom-Large-Precategory Y Z)
+      (f : hom-Large-Precategory X Y) →
+      ( comp-hom-Large-Precategory (comp-hom-Large-Precategory h g) f) ＝
+      ( comp-hom-Large-Precategory h (comp-hom-Large-Precategory g f))
+  associative-comp-hom-Large-Precategory h g f =
+    eq-involutive-eq
+      ( involutive-eq-associative-comp-hom-Large-Precategory h g f)
+
 open Large-Precategory public
+
+make-Large-Precategory :
+  {α : Level → Level} {β : Level → Level → Level}
+  ( obj : (l : Level) → UU (α l))
+  ( hom-set : {l1 l2 : Level} → obj l1 → obj l2 → Set (β l1 l2))
+  ( _∘_ :
+    {l1 l2 l3 : Level}
+    {X : obj l1} {Y : obj l2} {Z : obj l3} →
+    type-Set (hom-set Y Z) → type-Set (hom-set X Y) → type-Set (hom-set X Z))
+  ( id : {l : Level} {X : obj l} → type-Set (hom-set X X))
+  ( assoc-comp-hom :
+    {l1 l2 l3 l4 : Level}
+    {X : obj l1} {Y : obj l2} {Z : obj l3} {W : obj l4}
+    (h : type-Set (hom-set Z W))
+    (g : type-Set (hom-set Y Z))
+    (f : type-Set (hom-set X Y)) →
+    ( (h ∘ g) ∘ f) ＝ ( h ∘ (g ∘ f)))
+  ( left-unit-comp-hom :
+    {l1 l2 : Level} {X : obj l1} {Y : obj l2} (f : type-Set (hom-set X Y)) →
+    id ∘ f ＝ f)
+  ( right-unit-comp-hom :
+    {l1 l2 : Level} {X : obj l1} {Y : obj l2} (f : type-Set (hom-set X Y)) →
+    f ∘ id ＝ f) →
+  Large-Precategory α β
+make-Large-Precategory
+  obj hom-set _∘_ id assoc-comp-hom left-unit-comp-hom right-unit-comp-hom =
+  λ where
+    .obj-Large-Precategory → obj
+    .hom-set-Large-Precategory → hom-set
+    .comp-hom-Large-Precategory → _∘_
+    .id-hom-Large-Precategory → id
+    .involutive-eq-associative-comp-hom-Large-Precategory h g f →
+      involutive-eq-eq (assoc-comp-hom h g f)
+    .left-unit-law-comp-hom-Large-Precategory → left-unit-comp-hom
+    .right-unit-law-comp-hom-Large-Precategory → right-unit-comp-hom
+
+{-# INLINE make-Large-Precategory #-}
 ```
 
 ```agda
@@ -106,28 +162,13 @@ module _
   (C : Large-Precategory α β)
   where
 
-  hom-Large-Precategory :
-    {l1 l2 : Level}
-    (X : obj-Large-Precategory C l1)
-    (Y : obj-Large-Precategory C l2) →
-    UU (β l1 l2)
-  hom-Large-Precategory X Y = type-Set (hom-set-Large-Precategory C X Y)
-
-  is-set-hom-Large-Precategory :
-    {l1 l2 : Level}
-    (X : obj-Large-Precategory C l1)
-    (Y : obj-Large-Precategory C l2) →
-    is-set (hom-Large-Precategory X Y)
-  is-set-hom-Large-Precategory X Y =
-    is-set-type-Set (hom-set-Large-Precategory C X Y)
-
   ap-comp-hom-Large-Precategory :
     {l1 l2 l3 : Level}
     {X : obj-Large-Precategory C l1}
     {Y : obj-Large-Precategory C l2}
     {Z : obj-Large-Precategory C l3}
-    {g g' : hom-Large-Precategory Y Z} (p : g ＝ g')
-    {f f' : hom-Large-Precategory X Y} (q : f ＝ f') →
+    {g g' : hom-Large-Precategory C Y Z} (p : g ＝ g')
+    {f f' : hom-Large-Precategory C X Y} (q : f ＝ f') →
     comp-hom-Large-Precategory C g f ＝
     comp-hom-Large-Precategory C g' f'
   ap-comp-hom-Large-Precategory = ap-binary (comp-hom-Large-Precategory C)
@@ -137,9 +178,9 @@ module _
     {X : obj-Large-Precategory C l1}
     {Y : obj-Large-Precategory C l2}
     {Z : obj-Large-Precategory C l3} →
-    hom-Large-Precategory X Y →
-    hom-Large-Precategory Y Z →
-    hom-Large-Precategory X Z
+    hom-Large-Precategory C X Y →
+    hom-Large-Precategory C Y Z →
+    hom-Large-Precategory C X Z
   comp-hom-Large-Precategory' f g = comp-hom-Large-Precategory C g f
 ```
 
@@ -159,10 +200,8 @@ module _
     hom-set-Large-Precategory C
   pr1 (pr1 (pr2 (pr2 (precategory-Large-Precategory l)))) =
     comp-hom-Large-Precategory C
-  pr1 (pr2 (pr1 (pr2 (pr2 (precategory-Large-Precategory l)))) h g f) =
-    associative-comp-hom-Large-Precategory C h g f
-  pr2 (pr2 (pr1 (pr2 (pr2 (precategory-Large-Precategory l)))) h g f) =
-    inv-associative-comp-hom-Large-Precategory C h g f
+  pr2 (pr1 (pr2 (pr2 (precategory-Large-Precategory l)))) =
+    involutive-eq-associative-comp-hom-Large-Precategory C
   pr1 (pr2 (pr2 (pr2 (precategory-Large-Precategory l)))) x =
     id-hom-Large-Precategory C
   pr1 (pr2 (pr2 (pr2 (pr2 (precategory-Large-Precategory l))))) =
diff --git a/src/category-theory/large-subprecategories.lagda.md b/src/category-theory/large-subprecategories.lagda.md
index efeeebbc58..935a5ad81c 100644
--- a/src/category-theory/large-subprecategories.lagda.md
+++ b/src/category-theory/large-subprecategories.lagda.md
@@ -9,6 +9,10 @@ module category-theory.large-subprecategories where
 ```agda
 open import category-theory.large-precategories
 
+open import foundation.dependent-pair-types
+open import foundation.identity-types
+open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
 open import foundation.universe-levels
 ```
@@ -35,62 +39,196 @@ of the objects of `C` indexed by universe levels, and a family of subtypes `P₁
 of the morphisms of `C`, such that `P₁` contains all identity morphisms of
 objects in `P₀` and is closed under composition.
 
-## Definition
+## Definitions
 
 ### Large subprecategories
 
 ```agda
-module _
-  {α : Level → Level} {β : Level → Level → Level}
-  (γ : Level → Level) (δ : Level → Level → Level)
-  (C : Large-Precategory α β)
-  where
-
-  record
-    Large-Subprecategory : UUω
-    where
-    field
-      subtype-obj-Large-Subprecategory :
-        (l : Level) → subtype (γ l) (obj-Large-Precategory C l)
-      subtype-hom-Large-Subprecategory :
-        {l1 l2 : Level}
-        (X : obj-Large-Precategory C l1) (Y : obj-Large-Precategory C l2) →
-        is-in-subtype (subtype-obj-Large-Subprecategory l1) X →
-        is-in-subtype (subtype-obj-Large-Subprecategory l2) Y →
-        subtype (β l1 l2) (hom-Large-Precategory C X Y)
-      contains-id-Large-Subprecategory :
-        {l1 : Level} (X : obj-Large-Precategory C l1) →
-        (H : is-in-subtype (subtype-obj-Large-Subprecategory l1) X) →
-        is-in-subtype
-          ( subtype-hom-Large-Subprecategory X X H H)
-          ( id-hom-Large-Precategory C)
-      is-closed-under-composition-Large-Subprecategory :
-        {l1 l2 l3 : Level}
-        (X : obj-Large-Precategory C l1)
-        (Y : obj-Large-Precategory C l2)
-        (Z : obj-Large-Precategory C l3)
-        (g : hom-Large-Precategory C Y Z)
-        (f : hom-Large-Precategory C X Y) →
-        (K : is-in-subtype (subtype-obj-Large-Subprecategory l1) X) →
-        (L : is-in-subtype (subtype-obj-Large-Subprecategory l2) Y) →
-        (M : is-in-subtype (subtype-obj-Large-Subprecategory l3) Z) →
-        is-in-subtype (subtype-hom-Large-Subprecategory Y Z L M) g →
-        is-in-subtype (subtype-hom-Large-Subprecategory X Y K L) f →
-        is-in-subtype
-          ( subtype-hom-Large-Subprecategory X Z K M)
-          ( comp-hom-Large-Precategory C g f)
-
-  open Large-Subprecategory public
-
-module _
-  {α : Level → Level} {β : Level → Level → Level}
-  {γ : Level → Level} {δ : Level → Level → Level}
-  (C : Large-Precategory α β)
-  (P : Large-Subprecategory γ δ C)
+record
+  Large-Subprecategory
+    {α : Level → Level} {β : Level → Level → Level}
+    (γ : Level → Level) (δ : Level → Level → Level)
+    (C : Large-Precategory α β) : UUω
   where
+  field
+    subtype-obj-Large-Subprecategory :
+      (l : Level) → subtype (γ l) (obj-Large-Precategory C l)
 
   is-in-obj-Large-Subprecategory :
     {l : Level} → obj-Large-Precategory C l → UU (γ l)
-  is-in-obj-Large-Subprecategory =
-    is-in-subtype (subtype-obj-Large-Subprecategory P _)
+  is-in-obj-Large-Subprecategory {l} =
+    is-in-subtype (subtype-obj-Large-Subprecategory l)
+
+  obj-Large-Subprecategory :
+    (l : Level) → UU (α l ⊔ γ l)
+  obj-Large-Subprecategory l = type-subtype (subtype-obj-Large-Subprecategory l)
+
+  field
+    subtype-hom-Large-Subprecategory :
+      {l1 l2 : Level}
+      (X : obj-Large-Precategory C l1)
+      (Y : obj-Large-Precategory C l2) →
+      is-in-obj-Large-Subprecategory X →
+      is-in-obj-Large-Subprecategory Y →
+      subtype (δ l1 l2) (hom-Large-Precategory C X Y)
+
+  is-in-hom-is-in-obj-Large-Subprecategory :
+    {l1 l2 : Level}
+    {X : obj-Large-Precategory C l1}
+    {Y : obj-Large-Precategory C l2}
+    (x : is-in-obj-Large-Subprecategory X)
+    (y : is-in-obj-Large-Subprecategory Y) →
+    hom-Large-Precategory C X Y → UU (δ l1 l2)
+  is-in-hom-is-in-obj-Large-Subprecategory {l1} {l2} {X} {Y} x y =
+    is-in-subtype (subtype-hom-Large-Subprecategory X Y x y)
+
+  field
+    contains-id-Large-Subprecategory :
+      {l : Level} (X : obj-Large-Precategory C l) →
+      (H : is-in-obj-Large-Subprecategory X) →
+      is-in-hom-is-in-obj-Large-Subprecategory H H (id-hom-Large-Precategory C)
+
+    is-closed-under-composition-Large-Subprecategory :
+      {l1 l2 l3 : Level}
+      (X : obj-Large-Precategory C l1)
+      (Y : obj-Large-Precategory C l2)
+      (Z : obj-Large-Precategory C l3)
+      (g : hom-Large-Precategory C Y Z)
+      (f : hom-Large-Precategory C X Y) →
+      (K : is-in-obj-Large-Subprecategory X) →
+      (L : is-in-obj-Large-Subprecategory Y) →
+      (M : is-in-obj-Large-Subprecategory Z) →
+      is-in-hom-is-in-obj-Large-Subprecategory L M g →
+      is-in-hom-is-in-obj-Large-Subprecategory K L f →
+      is-in-hom-is-in-obj-Large-Subprecategory K M
+        ( comp-hom-Large-Precategory C g f)
+
+  hom-Large-Subprecategory :
+    {l1 l2 : Level}
+    (X : obj-Large-Subprecategory l1)
+    (Y : obj-Large-Subprecategory l2) → UU (β l1 l2 ⊔ δ l1 l2)
+  hom-Large-Subprecategory (X , x) (Y , y) =
+    type-subtype (subtype-hom-Large-Subprecategory X Y x y)
+
+  hom-set-Large-Subprecategory :
+    {l1 l2 : Level}
+    (X : obj-Large-Subprecategory l1)
+    (Y : obj-Large-Subprecategory l2) → Set (β l1 l2 ⊔ δ l1 l2)
+  hom-set-Large-Subprecategory (X , x) (Y , y) =
+    set-subset
+      ( hom-set-Large-Precategory C X Y)
+      ( subtype-hom-Large-Subprecategory X Y x y)
+
+  is-set-hom-Large-Subprecategory :
+    {l1 l2 : Level}
+    (X : obj-Large-Subprecategory l1)
+    (Y : obj-Large-Subprecategory l2) → is-set (hom-Large-Subprecategory X Y)
+  is-set-hom-Large-Subprecategory X Y =
+    is-set-type-Set (hom-set-Large-Subprecategory X Y)
+
+  id-hom-Large-Subprecategory :
+      {l : Level} (X : obj-Large-Subprecategory l) →
+      hom-Large-Subprecategory X X
+  id-hom-Large-Subprecategory (X , x) =
+    ( id-hom-Large-Precategory C , contains-id-Large-Subprecategory X x)
+
+  comp-hom-Large-Subprecategory :
+    {l1 l2 l3 : Level}
+    (X : obj-Large-Subprecategory l1)
+    (Y : obj-Large-Subprecategory l2)
+    (Z : obj-Large-Subprecategory l3) →
+    hom-Large-Subprecategory Y Z →
+    hom-Large-Subprecategory X Y →
+    hom-Large-Subprecategory X Z
+  comp-hom-Large-Subprecategory (X , x) (Y , y) (Z , z) (G , g) (F , f) =
+    ( comp-hom-Large-Precategory C G F ,
+      is-closed-under-composition-Large-Subprecategory X Y Z G F x y z g f)
+
+  associative-comp-hom-Large-Subprecategory :
+    {l1 l2 l3 l4 : Level}
+    (X : obj-Large-Subprecategory l1)
+    (Y : obj-Large-Subprecategory l2)
+    (Z : obj-Large-Subprecategory l3)
+    (W : obj-Large-Subprecategory l4)
+    (h : hom-Large-Subprecategory Z W)
+    (g : hom-Large-Subprecategory Y Z)
+    (f : hom-Large-Subprecategory X Y) →
+    comp-hom-Large-Subprecategory X Y W
+      ( comp-hom-Large-Subprecategory Y Z W h g)
+      ( f) ＝
+    comp-hom-Large-Subprecategory X Z W
+      ( h)
+      ( comp-hom-Large-Subprecategory X Y Z g f)
+  associative-comp-hom-Large-Subprecategory
+    ( X , x) (Y , y) (Z , z) (W , w) (H , h) (G , g) (F , f) =
+    eq-type-subtype
+      ( subtype-hom-Large-Subprecategory X W x w)
+      ( associative-comp-hom-Large-Precategory C H G F)
+
+  involutive-eq-associative-comp-hom-Large-Subprecategory :
+    {l1 l2 l3 l4 : Level}
+    (X : obj-Large-Subprecategory l1)
+    (Y : obj-Large-Subprecategory l2)
+    (Z : obj-Large-Subprecategory l3)
+    (W : obj-Large-Subprecategory l4)
+    (h : hom-Large-Subprecategory Z W)
+    (g : hom-Large-Subprecategory Y Z)
+    (f : hom-Large-Subprecategory X Y) →
+    comp-hom-Large-Subprecategory X Y W
+      ( comp-hom-Large-Subprecategory Y Z W h g)
+      ( f) ＝ⁱ
+    comp-hom-Large-Subprecategory X Z W
+      ( h)
+      ( comp-hom-Large-Subprecategory X Y Z g f)
+  involutive-eq-associative-comp-hom-Large-Subprecategory
+    X Y Z W h g f =
+    involutive-eq-eq (associative-comp-hom-Large-Subprecategory X Y Z W h g f)
+
+  left-unit-law-comp-hom-Large-Subprecategory :
+    {l1 l2 : Level}
+    (X : obj-Large-Subprecategory l1)
+    (Y : obj-Large-Subprecategory l2)
+    (f : hom-Large-Subprecategory X Y) →
+    comp-hom-Large-Subprecategory X Y Y (id-hom-Large-Subprecategory Y) f ＝ f
+  left-unit-law-comp-hom-Large-Subprecategory (X , x) (Y , y) (F , f) =
+    eq-type-subtype
+      ( subtype-hom-Large-Subprecategory X Y x y)
+      ( left-unit-law-comp-hom-Large-Precategory C F)
+
+  right-unit-law-comp-hom-Large-Subprecategory :
+    {l1 l2 : Level}
+    (X : obj-Large-Subprecategory l1)
+    (Y : obj-Large-Subprecategory l2)
+    (f : hom-Large-Subprecategory X Y) →
+    comp-hom-Large-Subprecategory X X Y f (id-hom-Large-Subprecategory X) ＝ f
+  right-unit-law-comp-hom-Large-Subprecategory (X , x) (Y , y) (F , f) =
+    eq-type-subtype
+      ( subtype-hom-Large-Subprecategory X Y x y)
+      ( right-unit-law-comp-hom-Large-Precategory C F)
+```
+
+### The underlying large precategory of a large subprecategory
+
+```agda
+  large-precategory-Large-Subprecategory :
+    Large-Precategory (λ l → α l ⊔ γ l) (λ l1 l2 → β l1 l2 ⊔ δ l1 l2)
+  large-precategory-Large-Subprecategory =
+    λ where
+      .obj-Large-Precategory →
+        obj-Large-Subprecategory
+      .hom-set-Large-Precategory →
+        hom-set-Large-Subprecategory
+      .comp-hom-Large-Precategory {X = X} {Y} {Z} →
+        comp-hom-Large-Subprecategory X Y Z
+      .id-hom-Large-Precategory {X = X} →
+        id-hom-Large-Subprecategory X
+      .involutive-eq-associative-comp-hom-Large-Precategory
+        {X = X} {Y} {Z} {W} →
+        involutive-eq-associative-comp-hom-Large-Subprecategory X Y Z W
+      .left-unit-law-comp-hom-Large-Precategory {X = X} {Y} →
+        left-unit-law-comp-hom-Large-Subprecategory X Y
+      .right-unit-law-comp-hom-Large-Precategory {X = X} {Y} →
+        right-unit-law-comp-hom-Large-Subprecategory X Y
+
+open Large-Subprecategory public
 ```
diff --git a/src/category-theory/natural-transformations-functors-from-small-to-large-precategories.lagda.md b/src/category-theory/natural-transformations-functors-from-small-to-large-precategories.lagda.md
index f76793468d..bde1b382d5 100644
--- a/src/category-theory/natural-transformations-functors-from-small-to-large-precategories.lagda.md
+++ b/src/category-theory/natural-transformations-functors-from-small-to-large-precategories.lagda.md
@@ -19,6 +19,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -262,7 +263,7 @@ module _
       ( map-functor-Small-Large-Precategory C D H)
       ( map-functor-Small-Large-Precategory C D I)
 
-  inv-associative-comp-natural-transformation-Small-Large-Precategory :
+  involutive-eq-associative-comp-natural-transformation-Small-Large-Precategory :
     {γF γG γH γI : Level}
     (F : functor-Small-Large-Precategory C D γF)
     (G : functor-Small-Large-Precategory C D γG)
@@ -271,12 +272,15 @@ module _
     (a : natural-transformation-Small-Large-Precategory C D F G)
     (b : natural-transformation-Small-Large-Precategory C D G H)
     (c : natural-transformation-Small-Large-Precategory C D H I) →
-    comp-natural-transformation-Small-Large-Precategory C D F H I c
-      ( comp-natural-transformation-Small-Large-Precategory C D F G H b a) ＝
     comp-natural-transformation-Small-Large-Precategory C D F G I
-      ( comp-natural-transformation-Small-Large-Precategory C D G H I c b) a
-  inv-associative-comp-natural-transformation-Small-Large-Precategory F G H I =
-    inv-associative-comp-natural-transformation-map-Small-Large-Precategory C D
+      ( comp-natural-transformation-Small-Large-Precategory C D G H I c b) a ＝ⁱ
+    comp-natural-transformation-Small-Large-Precategory C D F H I c
+      ( comp-natural-transformation-Small-Large-Precategory C D F G H b a)
+  involutive-eq-associative-comp-natural-transformation-Small-Large-Precategory
+    F G H I =
+    involutive-eq-associative-comp-natural-transformation-map-Small-Large-Precategory
+      ( C)
+      ( D)
       ( map-functor-Small-Large-Precategory C D F)
       ( map-functor-Small-Large-Precategory C D G)
       ( map-functor-Small-Large-Precategory C D H)
diff --git a/src/category-theory/natural-transformations-functors-precategories.lagda.md b/src/category-theory/natural-transformations-functors-precategories.lagda.md
index 2bb9c36a2c..75078505a6 100644
--- a/src/category-theory/natural-transformations-functors-precategories.lagda.md
+++ b/src/category-theory/natural-transformations-functors-precategories.lagda.md
@@ -21,6 +21,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -243,17 +244,17 @@ module _
       ( map-functor-Precategory C D H)
       ( map-functor-Precategory C D I)
 
-  inv-associative-comp-natural-transformation-Precategory :
+  involutive-eq-associative-comp-natural-transformation-Precategory :
     (F G H I : functor-Precategory C D)
     (α : natural-transformation-Precategory C D F G)
     (β : natural-transformation-Precategory C D G H)
     (γ : natural-transformation-Precategory C D H I) →
-    comp-natural-transformation-Precategory C D F H I γ
-      ( comp-natural-transformation-Precategory C D F G H β α) ＝
     comp-natural-transformation-Precategory C D F G I
-      ( comp-natural-transformation-Precategory C D G H I γ β) α
-  inv-associative-comp-natural-transformation-Precategory F G H I =
-    inv-associative-comp-natural-transformation-map-Precategory C D
+      ( comp-natural-transformation-Precategory C D G H I γ β) α ＝ⁱ
+    comp-natural-transformation-Precategory C D F H I γ
+      ( comp-natural-transformation-Precategory C D F G H β α)
+  involutive-eq-associative-comp-natural-transformation-Precategory F G H I =
+    involutive-eq-associative-comp-natural-transformation-map-Precategory C D
       ( map-functor-Precategory C D F)
       ( map-functor-Precategory C D G)
       ( map-functor-Precategory C D H)
diff --git a/src/category-theory/natural-transformations-maps-from-small-to-large-precategories.lagda.md b/src/category-theory/natural-transformations-maps-from-small-to-large-precategories.lagda.md
index 41fade4d73..3158288d63 100644
--- a/src/category-theory/natural-transformations-maps-from-small-to-large-precategories.lagda.md
+++ b/src/category-theory/natural-transformations-maps-from-small-to-large-precategories.lagda.md
@@ -22,6 +22,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
 open import foundation.universe-levels
 ```
@@ -343,7 +344,7 @@ module _
           ( hom-natural-transformation-map-Small-Large-Precategory
             C D F G a x))
 
-  inv-associative-comp-natural-transformation-map-Small-Large-Precategory :
+  involutive-eq-associative-comp-natural-transformation-map-Small-Large-Precategory :
     {γF γG γH γI : Level}
     (F : map-Small-Large-Precategory C D γF)
     (G : map-Small-Large-Precategory C D γG)
@@ -352,22 +353,14 @@ module _
     (a : natural-transformation-map-Small-Large-Precategory C D F G)
     (b : natural-transformation-map-Small-Large-Precategory C D G H)
     (c : natural-transformation-map-Small-Large-Precategory C D H I) →
-    comp-natural-transformation-map-Small-Large-Precategory C D F H I c
-      ( comp-natural-transformation-map-Small-Large-Precategory
-        C D F G H b a) ＝
     comp-natural-transformation-map-Small-Large-Precategory C D F G I
       ( comp-natural-transformation-map-Small-Large-Precategory C D G H I c b)
-      ( a)
-  inv-associative-comp-natural-transformation-map-Small-Large-Precategory
+      ( a) ＝ⁱ
+    comp-natural-transformation-map-Small-Large-Precategory C D F H I c
+      ( comp-natural-transformation-map-Small-Large-Precategory C D F G H b a)
+  involutive-eq-associative-comp-natural-transformation-map-Small-Large-Precategory
     F G H I a b c =
-    eq-htpy-hom-natural-transformation-map-Small-Large-Precategory
-      C D F I _ _
-      ( λ x →
-        inv-associative-comp-hom-Large-Precategory D
-          ( hom-natural-transformation-map-Small-Large-Precategory
-            C D H I c x)
-          ( hom-natural-transformation-map-Small-Large-Precategory
-            C D G H b x)
-          ( hom-natural-transformation-map-Small-Large-Precategory
-            C D F G a x))
+    involutive-eq-eq
+      ( associative-comp-natural-transformation-map-Small-Large-Precategory
+          F G H I a b c)
 ```
diff --git a/src/category-theory/natural-transformations-maps-precategories.lagda.md b/src/category-theory/natural-transformations-maps-precategories.lagda.md
index 488f1fbf3e..757216ed17 100644
--- a/src/category-theory/natural-transformations-maps-precategories.lagda.md
+++ b/src/category-theory/natural-transformations-maps-precategories.lagda.md
@@ -21,6 +21,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
 open import foundation.universe-levels
 ```
@@ -294,20 +295,17 @@ module _
         ( hom-family-natural-transformation-map-Precategory C D G H β x)
         ( hom-family-natural-transformation-map-Precategory C D F G α x))
 
-  inv-associative-comp-natural-transformation-map-Precategory :
+  involutive-eq-associative-comp-natural-transformation-map-Precategory :
     (F G H I : map-Precategory C D)
     (α : natural-transformation-map-Precategory C D F G)
     (β : natural-transformation-map-Precategory C D G H)
     (γ : natural-transformation-map-Precategory C D H I) →
-    comp-natural-transformation-map-Precategory C D F H I γ
-      ( comp-natural-transformation-map-Precategory C D F G H β α) ＝
     comp-natural-transformation-map-Precategory C D F G I
-      ( comp-natural-transformation-map-Precategory C D G H I γ β) α
-  inv-associative-comp-natural-transformation-map-Precategory F G H I α β γ =
-    eq-htpy-hom-family-natural-transformation-map-Precategory C D F I _ _
-    ( λ x →
-      inv-associative-comp-hom-Precategory D
-        ( hom-family-natural-transformation-map-Precategory C D H I γ x)
-        ( hom-family-natural-transformation-map-Precategory C D G H β x)
-        ( hom-family-natural-transformation-map-Precategory C D F G α x))
+      ( comp-natural-transformation-map-Precategory C D G H I γ β) α ＝ⁱ
+    comp-natural-transformation-map-Precategory C D F H I γ
+      ( comp-natural-transformation-map-Precategory C D F G H β α)
+  involutive-eq-associative-comp-natural-transformation-map-Precategory
+    F G H I α β γ =
+    involutive-eq-eq
+      ( associative-comp-natural-transformation-map-Precategory F G H I α β γ)
 ```
diff --git a/src/category-theory/nonunital-precategories.lagda.md b/src/category-theory/nonunital-precategories.lagda.md
index 21f5ae93c3..32f4782eab 100644
--- a/src/category-theory/nonunital-precategories.lagda.md
+++ b/src/category-theory/nonunital-precategories.lagda.md
@@ -15,6 +15,7 @@ open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.universe-levels
@@ -24,10 +25,15 @@ open import foundation.universe-levels
 
 ## Idea
 
-A **nonunital precategory** is a [precategory](category-theory.precategories.md)
-that may not have identity maps. In other words, it is an associative
+A {{#concept "nonunital precategory" Agda=Nonunital-Precategory}} is a
+[precategory](category-theory.precategories.md) that may not have identity
+morphisms. In other words, it is an associative
 [composition operation on binary families of sets](category-theory.composition-operations-on-binary-families-of-sets.md).
-Such an object may also be called a **semiprecategory**.
+Such a structure may also be referred to as a _semiprecategory_.
+
+Perhaps surprisingly, there is [at most one](foundation.subterminal-types.md)
+way to equip nonunital precategories with identity morphisms, so precategories
+form a [subtype](foundation-core.subtypes.md) of nonunital precategories.
 
 ## Definition
 
@@ -94,15 +100,15 @@ module _
       ( hom-set-Nonunital-Precategory)
       ( associative-composition-operation-Nonunital-Precategory)
 
-  inv-associative-comp-hom-Nonunital-Precategory :
+  involutive-eq-associative-comp-hom-Nonunital-Precategory :
     {x y z w : obj-Nonunital-Precategory}
     (h : hom-Nonunital-Precategory z w)
     (g : hom-Nonunital-Precategory y z)
     (f : hom-Nonunital-Precategory x y) →
-    comp-hom-Nonunital-Precategory h (comp-hom-Nonunital-Precategory g f) ＝
-    comp-hom-Nonunital-Precategory (comp-hom-Nonunital-Precategory h g) f
-  inv-associative-comp-hom-Nonunital-Precategory =
-    inv-witness-associative-composition-operation-binary-family-Set
+    comp-hom-Nonunital-Precategory (comp-hom-Nonunital-Precategory h g) f ＝ⁱ
+    comp-hom-Nonunital-Precategory h (comp-hom-Nonunital-Precategory g f)
+  involutive-eq-associative-comp-hom-Nonunital-Precategory =
+    involutive-eq-associative-composition-operation-binary-family-Set
       ( hom-set-Nonunital-Precategory)
       ( associative-composition-operation-Nonunital-Precategory)
 ```
diff --git a/src/category-theory/one-object-precategories.lagda.md b/src/category-theory/one-object-precategories.lagda.md
index 376efd5dee..76dbd98741 100644
--- a/src/category-theory/one-object-precategories.lagda.md
+++ b/src/category-theory/one-object-precategories.lagda.md
@@ -7,7 +7,6 @@ module category-theory.one-object-precategories where
 <details><summary>Imports</summary>
 
 ```agda
-open import category-theory.composition-operations-on-binary-families-of-sets
 open import category-theory.endomorphisms-in-precategories
 open import category-theory.precategories
 
@@ -64,7 +63,7 @@ module _
 
   hom-set-one-object-precategory-Monoid :
     unit → unit → Set l
-  hom-set-one-object-precategory-Monoid star star = set-Monoid M
+  hom-set-one-object-precategory-Monoid _ _ = set-Monoid M
 
   hom-one-object-precategory-Monoid :
     unit → unit → UU l
@@ -76,7 +75,7 @@ module _
     hom-one-object-precategory-Monoid y z →
     hom-one-object-precategory-Monoid x y →
     hom-one-object-precategory-Monoid x z
-  comp-hom-one-object-precategory-Monoid {star} {star} {star} =
+  comp-hom-one-object-precategory-Monoid =
     mul-Monoid M
 
   associative-comp-hom-one-object-precategory-Monoid :
@@ -90,24 +89,12 @@ module _
     comp-hom-one-object-precategory-Monoid
       ( h)
       ( comp-hom-one-object-precategory-Monoid g f)
-  associative-comp-hom-one-object-precategory-Monoid
-    {star} {star} {star} {star} =
+  associative-comp-hom-one-object-precategory-Monoid =
     associative-mul-Monoid M
 
-  associative-composition-operation-one-object-precategory-Monoid :
-    associative-composition-operation-binary-family-Set
-      hom-set-one-object-precategory-Monoid
-  pr1 associative-composition-operation-one-object-precategory-Monoid =
-    comp-hom-one-object-precategory-Monoid
-  pr2 associative-composition-operation-one-object-precategory-Monoid =
-    is-associative-witness-associative-composition-operation-binary-family-Set
-      ( hom-set-one-object-precategory-Monoid)
-      ( comp-hom-one-object-precategory-Monoid)
-      ( associative-comp-hom-one-object-precategory-Monoid)
-
   id-hom-one-object-precategory-Monoid :
     (x : unit) → hom-one-object-precategory-Monoid x x
-  id-hom-one-object-precategory-Monoid star = unit-Monoid M
+  id-hom-one-object-precategory-Monoid _ = unit-Monoid M
 
   left-unit-law-comp-hom-one-object-precategory-Monoid :
     {x y : unit} (f : hom-one-object-precategory-Monoid x y) →
@@ -115,7 +102,7 @@ module _
       ( id-hom-one-object-precategory-Monoid y)
       ( f) ＝
     f
-  left-unit-law-comp-hom-one-object-precategory-Monoid {star} {star} =
+  left-unit-law-comp-hom-one-object-precategory-Monoid =
     left-unit-law-mul-Monoid M
 
   right-unit-law-comp-hom-one-object-precategory-Monoid :
@@ -124,28 +111,19 @@ module _
       ( f)
       ( id-hom-one-object-precategory-Monoid x) ＝
     f
-  right-unit-law-comp-hom-one-object-precategory-Monoid {star} {star} =
+  right-unit-law-comp-hom-one-object-precategory-Monoid =
     right-unit-law-mul-Monoid M
 
-  is-unital-composition-operation-one-object-precategory-Monoid :
-    is-unital-composition-operation-binary-family-Set
-      hom-set-one-object-precategory-Monoid
-      comp-hom-one-object-precategory-Monoid
-  pr1 is-unital-composition-operation-one-object-precategory-Monoid =
-    id-hom-one-object-precategory-Monoid
-  pr1 (pr2 is-unital-composition-operation-one-object-precategory-Monoid) =
-    left-unit-law-comp-hom-one-object-precategory-Monoid
-  pr2 (pr2 is-unital-composition-operation-one-object-precategory-Monoid) =
-    right-unit-law-comp-hom-one-object-precategory-Monoid
-
   precategory-one-object-precategory-Monoid : Precategory lzero l
-  pr1 precategory-one-object-precategory-Monoid = unit
-  pr1 (pr2 precategory-one-object-precategory-Monoid) =
-    hom-set-one-object-precategory-Monoid
-  pr1 (pr2 (pr2 precategory-one-object-precategory-Monoid)) =
-    associative-composition-operation-one-object-precategory-Monoid
-  pr2 (pr2 (pr2 precategory-one-object-precategory-Monoid)) =
-    is-unital-composition-operation-one-object-precategory-Monoid
+  precategory-one-object-precategory-Monoid =
+    make-Precategory
+      ( unit)
+      ( hom-set-one-object-precategory-Monoid)
+      ( comp-hom-one-object-precategory-Monoid)
+      ( id-hom-one-object-precategory-Monoid)
+      ( associative-comp-hom-one-object-precategory-Monoid)
+      ( left-unit-law-comp-hom-one-object-precategory-Monoid)
+      ( right-unit-law-comp-hom-one-object-precategory-Monoid)
 
   one-object-precategory-Monoid : One-Object-Precategory lzero l
   pr1 one-object-precategory-Monoid = precategory-one-object-precategory-Monoid
diff --git a/src/category-theory/opposite-categories.lagda.md b/src/category-theory/opposite-categories.lagda.md
index 45be60417f..8eefb6735c 100644
--- a/src/category-theory/opposite-categories.lagda.md
+++ b/src/category-theory/opposite-categories.lagda.md
@@ -17,8 +17,8 @@ open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.involutions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
@@ -96,15 +96,16 @@ module _
   associative-comp-hom-opposite-Category =
     associative-comp-hom-opposite-Precategory (precategory-Category C)
 
-  inv-associative-comp-hom-opposite-Category :
+  involutive-eq-associative-comp-hom-opposite-Category :
     {x y z w : obj-opposite-Category}
     (h : hom-opposite-Category z w)
     (g : hom-opposite-Category y z)
     (f : hom-opposite-Category x y) →
-    comp-hom-opposite-Category h (comp-hom-opposite-Category g f) ＝
-    comp-hom-opposite-Category (comp-hom-opposite-Category h g) f
-  inv-associative-comp-hom-opposite-Category =
-    inv-associative-comp-hom-opposite-Precategory (precategory-Category C)
+    comp-hom-opposite-Category (comp-hom-opposite-Category h g) f ＝ⁱ
+    comp-hom-opposite-Category h (comp-hom-opposite-Category g f)
+  involutive-eq-associative-comp-hom-opposite-Category =
+    involutive-eq-associative-comp-hom-opposite-Precategory
+      ( precategory-Category C)
 
   id-hom-opposite-Category :
     {x : obj-opposite-Category} → hom-opposite-Category x x
diff --git a/src/category-theory/opposite-large-precategories.lagda.md b/src/category-theory/opposite-large-precategories.lagda.md
index 817c81bc5d..3365e3523b 100644
--- a/src/category-theory/opposite-large-precategories.lagda.md
+++ b/src/category-theory/opposite-large-precategories.lagda.md
@@ -7,18 +7,15 @@ module category-theory.opposite-large-precategories where
 <details><summary>Imports</summary>
 
 ```agda
-open import category-theory.composition-operations-on-binary-families-of-sets
 open import category-theory.isomorphisms-in-large-precategories
 open import category-theory.large-precategories
 
 open import foundation.dependent-pair-types
-open import foundation.equality-dependent-pair-types
 open import foundation.equivalences
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.involutions
-open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -64,7 +61,7 @@ module _
     hom-opposite-Large-Precategory X Z
   comp-hom-opposite-Large-Precategory g f = comp-hom-Large-Precategory C f g
 
-  associative-comp-hom-opposite-Large-Precategory :
+  involutive-eq-associative-comp-hom-opposite-Large-Precategory :
     {l1 l2 l3 l4 : Level}
     {X : obj-opposite-Large-Precategory l1}
     {Y : obj-opposite-Large-Precategory l2}
@@ -75,14 +72,14 @@ module _
     (f : hom-opposite-Large-Precategory X Y) →
     comp-hom-opposite-Large-Precategory
       ( comp-hom-opposite-Large-Precategory h g)
-      ( f) ＝
+      ( f) ＝ⁱ
     comp-hom-opposite-Large-Precategory
       ( h)
       ( comp-hom-opposite-Large-Precategory g f)
-  associative-comp-hom-opposite-Large-Precategory h g f =
-    inv-associative-comp-hom-Large-Precategory C f g h
+  involutive-eq-associative-comp-hom-opposite-Large-Precategory h g f =
+    invⁱ (involutive-eq-associative-comp-hom-Large-Precategory C f g h)
 
-  inv-associative-comp-hom-opposite-Large-Precategory :
+  associative-comp-hom-opposite-Large-Precategory :
     {l1 l2 l3 l4 : Level}
     {X : obj-opposite-Large-Precategory l1}
     {Y : obj-opposite-Large-Precategory l2}
@@ -91,14 +88,15 @@ module _
     (h : hom-opposite-Large-Precategory Z W)
     (g : hom-opposite-Large-Precategory Y Z)
     (f : hom-opposite-Large-Precategory X Y) →
-    comp-hom-opposite-Large-Precategory
-      ( h)
-      ( comp-hom-opposite-Large-Precategory g f) ＝
     comp-hom-opposite-Large-Precategory
       ( comp-hom-opposite-Large-Precategory h g)
-      ( f)
-  inv-associative-comp-hom-opposite-Large-Precategory h g f =
-    associative-comp-hom-Large-Precategory C f g h
+      ( f) ＝
+    comp-hom-opposite-Large-Precategory
+      ( h)
+      ( comp-hom-opposite-Large-Precategory g f)
+  associative-comp-hom-opposite-Large-Precategory h g f =
+    eq-involutive-eq
+      ( involutive-eq-associative-comp-hom-opposite-Large-Precategory h g f)
 
   id-hom-opposite-Large-Precategory :
     {l1 : Level} {X : obj-opposite-Large-Precategory l1} →
@@ -132,10 +130,9 @@ module _
     comp-hom-opposite-Large-Precategory
   id-hom-Large-Precategory opposite-Large-Precategory =
     id-hom-opposite-Large-Precategory
-  associative-comp-hom-Large-Precategory opposite-Large-Precategory =
-    associative-comp-hom-opposite-Large-Precategory
-  inv-associative-comp-hom-Large-Precategory opposite-Large-Precategory =
-    inv-associative-comp-hom-opposite-Large-Precategory
+  involutive-eq-associative-comp-hom-Large-Precategory
+    opposite-Large-Precategory =
+    involutive-eq-associative-comp-hom-opposite-Large-Precategory
   left-unit-law-comp-hom-Large-Precategory opposite-Large-Precategory =
     left-unit-law-comp-hom-opposite-Large-Precategory
   right-unit-law-comp-hom-Large-Precategory opposite-Large-Precategory =
diff --git a/src/category-theory/opposite-precategories.lagda.md b/src/category-theory/opposite-precategories.lagda.md
index 142ddf7f2f..d349bfe93e 100644
--- a/src/category-theory/opposite-precategories.lagda.md
+++ b/src/category-theory/opposite-precategories.lagda.md
@@ -7,18 +7,16 @@ module category-theory.opposite-precategories where
 <details><summary>Imports</summary>
 
 ```agda
-open import category-theory.composition-operations-on-binary-families-of-sets
 open import category-theory.isomorphisms-in-precategories
 open import category-theory.precategories
 
 open import foundation.dependent-pair-types
-open import foundation.equality-dependent-pair-types
 open import foundation.equivalences
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.involutions
-open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -54,35 +52,26 @@ module _
     hom-opposite-Precategory x z
   comp-hom-opposite-Precategory g f = comp-hom-Precategory C f g
 
-  associative-comp-hom-opposite-Precategory :
+  involutive-eq-associative-comp-hom-opposite-Precategory :
     {x y z w : obj-opposite-Precategory}
     (h : hom-opposite-Precategory z w)
     (g : hom-opposite-Precategory y z)
     (f : hom-opposite-Precategory x y) →
-    ( comp-hom-opposite-Precategory (comp-hom-opposite-Precategory h g) f) ＝
+    ( comp-hom-opposite-Precategory (comp-hom-opposite-Precategory h g) f) ＝ⁱ
     ( comp-hom-opposite-Precategory h (comp-hom-opposite-Precategory g f))
-  associative-comp-hom-opposite-Precategory h g f =
-    inv-witness-associative-composition-operation-binary-family-Set
-      ( hom-set-Precategory C)
-      ( associative-composition-operation-Precategory C)
-      ( f)
-      ( g)
-      ( h)
-
-  inv-associative-comp-hom-opposite-Precategory :
+  involutive-eq-associative-comp-hom-opposite-Precategory h g f =
+    invⁱ (involutive-eq-associative-comp-hom-Precategory C f g h)
+
+  associative-comp-hom-opposite-Precategory :
     {x y z w : obj-opposite-Precategory}
     (h : hom-opposite-Precategory z w)
     (g : hom-opposite-Precategory y z)
     (f : hom-opposite-Precategory x y) →
-    ( comp-hom-opposite-Precategory h (comp-hom-opposite-Precategory g f)) ＝
-    ( comp-hom-opposite-Precategory (comp-hom-opposite-Precategory h g) f)
-  inv-associative-comp-hom-opposite-Precategory h g f =
-    witness-associative-composition-operation-binary-family-Set
-      ( hom-set-Precategory C)
-      ( associative-composition-operation-Precategory C)
-      ( f)
-      ( g)
-      ( h)
+    ( comp-hom-opposite-Precategory (comp-hom-opposite-Precategory h g) f) ＝
+    ( comp-hom-opposite-Precategory h (comp-hom-opposite-Precategory g f))
+  associative-comp-hom-opposite-Precategory h g f =
+    eq-involutive-eq
+      ( involutive-eq-associative-comp-hom-opposite-Precategory h g f)
 
   id-hom-opposite-Precategory :
     {x : obj-opposite-Precategory} → hom-opposite-Precategory x x
@@ -105,10 +94,8 @@ module _
   pr1 opposite-Precategory = obj-opposite-Precategory
   pr1 (pr2 opposite-Precategory) = hom-set-opposite-Precategory
   pr1 (pr1 (pr2 (pr2 opposite-Precategory))) = comp-hom-opposite-Precategory
-  pr1 (pr2 (pr1 (pr2 (pr2 opposite-Precategory))) h g f) =
-    associative-comp-hom-opposite-Precategory h g f
-  pr2 (pr2 (pr1 (pr2 (pr2 opposite-Precategory))) h g f) =
-    inv-associative-comp-hom-opposite-Precategory h g f
+  pr2 (pr1 (pr2 (pr2 opposite-Precategory))) =
+    involutive-eq-associative-comp-hom-opposite-Precategory
   pr1 (pr2 (pr2 (pr2 opposite-Precategory))) x = id-hom-opposite-Precategory {x}
   pr1 (pr2 (pr2 (pr2 (pr2 opposite-Precategory)))) =
     left-unit-law-comp-hom-opposite-Precategory
diff --git a/src/category-theory/opposite-preunivalent-categories.lagda.md b/src/category-theory/opposite-preunivalent-categories.lagda.md
index 18f52b4ff6..4ea17ae265 100644
--- a/src/category-theory/opposite-preunivalent-categories.lagda.md
+++ b/src/category-theory/opposite-preunivalent-categories.lagda.md
@@ -18,8 +18,8 @@ open import foundation.equivalences
 open import foundation.identity-types
 open import foundation.involutions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
@@ -94,34 +94,34 @@ module _
   comp-hom-opposite-Preunivalent-Category =
     comp-hom-opposite-Precategory (precategory-Preunivalent-Category C)
 
-  associative-comp-hom-opposite-Preunivalent-Category :
+  involutive-eq-associative-comp-hom-opposite-Preunivalent-Category :
     {x y z w : obj-opposite-Preunivalent-Category}
     (h : hom-opposite-Preunivalent-Category z w)
     (g : hom-opposite-Preunivalent-Category y z)
     (f : hom-opposite-Preunivalent-Category x y) →
     comp-hom-opposite-Preunivalent-Category
       ( comp-hom-opposite-Preunivalent-Category h g)
-      ( f) ＝
+      ( f) ＝ⁱ
     comp-hom-opposite-Preunivalent-Category
       ( h)
       ( comp-hom-opposite-Preunivalent-Category g f)
-  associative-comp-hom-opposite-Preunivalent-Category =
-    associative-comp-hom-opposite-Precategory
+  involutive-eq-associative-comp-hom-opposite-Preunivalent-Category =
+    involutive-eq-associative-comp-hom-opposite-Precategory
       ( precategory-Preunivalent-Category C)
 
-  inv-associative-comp-hom-opposite-Preunivalent-Category :
+  associative-comp-hom-opposite-Preunivalent-Category :
     {x y z w : obj-opposite-Preunivalent-Category}
     (h : hom-opposite-Preunivalent-Category z w)
     (g : hom-opposite-Preunivalent-Category y z)
     (f : hom-opposite-Preunivalent-Category x y) →
-    comp-hom-opposite-Preunivalent-Category
-      ( h)
-      ( comp-hom-opposite-Preunivalent-Category g f) ＝
     comp-hom-opposite-Preunivalent-Category
       ( comp-hom-opposite-Preunivalent-Category h g)
-      ( f)
-  inv-associative-comp-hom-opposite-Preunivalent-Category =
-    inv-associative-comp-hom-opposite-Precategory
+      ( f) ＝
+    comp-hom-opposite-Preunivalent-Category
+      ( h)
+      ( comp-hom-opposite-Preunivalent-Category g f)
+  associative-comp-hom-opposite-Preunivalent-Category =
+    associative-comp-hom-opposite-Precategory
       ( precategory-Preunivalent-Category C)
 
   id-hom-opposite-Preunivalent-Category :
diff --git a/src/category-theory/precategories.lagda.md b/src/category-theory/precategories.lagda.md
index de63ac4bab..782ff09569 100644
--- a/src/category-theory/precategories.lagda.md
+++ b/src/category-theory/precategories.lagda.md
@@ -17,6 +17,7 @@ open import foundation.function-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.truncated-types
 open import foundation.truncation-levels
 open import foundation.universe-levels
@@ -26,21 +27,38 @@ open import foundation.universe-levels
 
 ## Idea
 
-A **precategory** in Homotopy Type Theory consists of:
-
-- a type `A` of objects,
-- for each pair of objects `x y : A`, a [set](foundation-core.sets.md) of
-  morphisms `hom x y : Set`, together with a composition operation
-  `_∘_ : hom y z → hom x y → hom x z` such that:
-- `(h ∘ g) ∘ f = h ∘ (g ∘ f)` for any morphisms `h : hom z w`, `g : hom y z` and
-  `f : hom x y`,
-- for each object `x : A` there is a morphism `id_x : hom x x` such that
-  `id_x ∘ f = f` and `g ∘ id_x = g` for any morphisms `f : hom x y` and
-  `g : hom z x`.
-
-The reason this is called a *pre*category and not a category in Homotopy Type
-Theory is that we want to reserve that name for precategories where the
-identities between the objects are exactly the isomorphisms.
+A {{#concept "precategory" Agda=Precategory}} `𝒞` in Homotopy Type Theory is the
+structure of an associative and unital
+[composition operation](category-theory.composition-operations-on-binary-families-of-sets.md)
+on a binary familiy of sets.
+
+This means a precategory consists of:
+
+- **Objects.** A type `Ob 𝒞` of _objects_.
+- **Morphisms.** For each pair of objects `x y : Ob 𝒞`, a
+  [set](foundation-core.sets.md) of _morphisms_ `hom 𝒞 x y : Set`.
+- **Composition.** For every triple of objects `x y z : Ob 𝒞` there is a
+  _composition operation_ on morphisms
+  ```text
+    _∘_ : hom 𝒞 y z → hom 𝒞 x y → hom 𝒞 x z.
+  ```
+- **Associativity.** For every triple of composable morphisms, we have
+  ```text
+    (h ∘ g) ∘ f ＝ h ∘ (g ∘ f).
+  ```
+- **Identity morphisms.** For every object `x : Ob 𝒞`, there is a distinguished
+  _identity_ morphism `id_x : hom 𝒞 x x`.
+- **Unitality.** The identity morphisms are two-sided units for the composition
+  operation, meaning that for every `f : hom 𝒞 x y` we have
+  ```text
+    id_y ∘ f ＝ f   and   f ∘ id_x ＝ f.
+  ```
+
+**Note.** The reason this is called a *pre*category and not a _category_ in
+Homotopy Type Theory is that we reserve that name for precategories where the
+[identity types](foundation-core.identity-types.md) of the type of objects are
+characterized by the
+[isomorphism sets](category-theory.isomorphisms-in-precategories.md).
 
 ## Definitions
 
@@ -88,6 +106,34 @@ Precategory l1 l2 =
                 ( hom-set)
                 ( comp-hom))))
 
+make-Precategory :
+  { l1 l2 : Level}
+  ( obj : UU l1)
+  ( hom-set : obj → obj → Set l2)
+  ( _∘_ : composition-operation-binary-family-Set hom-set)
+  ( id : (x : obj) → type-Set (hom-set x x))
+  ( assoc-comp-hom :
+    { x y z w : obj} →
+    ( h : type-Set (hom-set z w))
+    ( g : type-Set (hom-set y z))
+    ( f : type-Set (hom-set x y)) →
+    ( (h ∘ g) ∘ f ＝ h ∘ (g ∘ f)))
+  ( left-unit-comp-hom :
+    { x y : obj} (f : type-Set (hom-set x y)) → id y ∘ f ＝ f)
+  ( right-unit-comp-hom :
+    { x y : obj} (f : type-Set (hom-set x y)) → f ∘ id x ＝ f) →
+  Precategory l1 l2
+make-Precategory
+  obj hom-set _∘_ id assoc-comp-hom left-unit-comp-hom right-unit-comp-hom =
+  ( ( obj) ,
+    ( hom-set) ,
+    ( _∘_ , (λ h g f → involutive-eq-eq (assoc-comp-hom h g f))) ,
+    ( id) ,
+    ( left-unit-comp-hom) ,
+    ( right-unit-comp-hom))
+
+{-# INLINE make-Precategory #-}
+
 module _
   {l1 l2 : Level} (C : Precategory l1 l2)
   where
@@ -126,27 +172,27 @@ module _
     hom-Precategory x z
   comp-hom-Precategory' f g = comp-hom-Precategory g f
 
-  associative-comp-hom-Precategory :
+  involutive-eq-associative-comp-hom-Precategory :
     {x y z w : obj-Precategory}
     (h : hom-Precategory z w)
     (g : hom-Precategory y z)
     (f : hom-Precategory x y) →
-    ( comp-hom-Precategory (comp-hom-Precategory h g) f) ＝
+    ( comp-hom-Precategory (comp-hom-Precategory h g) f) ＝ⁱ
     ( comp-hom-Precategory h (comp-hom-Precategory g f))
-  associative-comp-hom-Precategory =
-    witness-associative-composition-operation-binary-family-Set
+  involutive-eq-associative-comp-hom-Precategory =
+    involutive-eq-associative-composition-operation-binary-family-Set
       ( hom-set-Precategory)
       ( associative-composition-operation-Precategory)
 
-  inv-associative-comp-hom-Precategory :
+  associative-comp-hom-Precategory :
     {x y z w : obj-Precategory}
     (h : hom-Precategory z w)
     (g : hom-Precategory y z)
     (f : hom-Precategory x y) →
-    ( comp-hom-Precategory h (comp-hom-Precategory g f)) ＝
-    ( comp-hom-Precategory (comp-hom-Precategory h g) f)
-  inv-associative-comp-hom-Precategory =
-    inv-witness-associative-composition-operation-binary-family-Set
+    ( comp-hom-Precategory (comp-hom-Precategory h g) f) ＝
+    ( comp-hom-Precategory h (comp-hom-Precategory g f))
+  associative-comp-hom-Precategory =
+    witness-associative-composition-operation-binary-family-Set
       ( hom-set-Precategory)
       ( associative-composition-operation-Precategory)
 
@@ -276,7 +322,7 @@ module _
 - [Functors between precategories](category-theory.categories.md) are
   [structure](foundation.structure.md)-preserving maps of precategories.
 - [Large precategories](category-theory.large-precategories.md) are
-  precategories whose collection of objects form a large type.
+  precategories whose collections of objects and morphisms form large types.
 
 ## External links
 
diff --git a/src/category-theory/precategory-of-elements-of-a-presheaf.lagda.md b/src/category-theory/precategory-of-elements-of-a-presheaf.lagda.md
index 41a42dec87..2e36074e3f 100644
--- a/src/category-theory/precategory-of-elements-of-a-presheaf.lagda.md
+++ b/src/category-theory/precategory-of-elements-of-a-presheaf.lagda.md
@@ -15,7 +15,6 @@ open import category-theory.presheaf-categories
 open import foundation.action-on-identifications-functions
 open import foundation.category-of-sets
 open import foundation.dependent-pair-types
-open import foundation.function-extensionality
 open import foundation.identity-types
 open import foundation.sets
 open import foundation.subtypes
@@ -109,27 +108,6 @@ module _
         ( comp-hom-precategory-of-elements-presheaf-Precategory g f))
       ( associative-comp-hom-Precategory C (pr1 h) (pr1 g) (pr1 f))
 
-  inv-associative-comp-hom-precategory-of-elements-presheaf-Precategory :
-    {X Y Z W : obj-precategory-of-elements-presheaf-Precategory} →
-    (h : hom-precategory-of-elements-presheaf-Precategory Z W)
-    (g : hom-precategory-of-elements-presheaf-Precategory Y Z)
-    (f : hom-precategory-of-elements-presheaf-Precategory X Y) →
-    comp-hom-precategory-of-elements-presheaf-Precategory
-      ( h)
-      ( comp-hom-precategory-of-elements-presheaf-Precategory g f) ＝
-    comp-hom-precategory-of-elements-presheaf-Precategory
-      ( comp-hom-precategory-of-elements-presheaf-Precategory h g)
-      ( f)
-  inv-associative-comp-hom-precategory-of-elements-presheaf-Precategory h g f =
-    eq-hom-precategory-of-elements-presheaf-Precategory
-      ( comp-hom-precategory-of-elements-presheaf-Precategory
-        ( h)
-        ( comp-hom-precategory-of-elements-presheaf-Precategory g f))
-      ( comp-hom-precategory-of-elements-presheaf-Precategory
-        ( comp-hom-precategory-of-elements-presheaf-Precategory h g)
-        ( f))
-      ( inv-associative-comp-hom-Precategory C (pr1 h) (pr1 g) (pr1 f))
-
   id-hom-precategory-of-elements-presheaf-Precategory :
     {X : obj-precategory-of-elements-presheaf-Precategory} →
     hom-precategory-of-elements-presheaf-Precategory X X
@@ -169,24 +147,15 @@ module _
       ( right-unit-law-comp-hom-Precategory C (pr1 f))
 
   precategory-of-elements-presheaf-Precategory : Precategory (l1 ⊔ l3) (l2 ⊔ l3)
-  pr1 precategory-of-elements-presheaf-Precategory =
-    obj-precategory-of-elements-presheaf-Precategory
-  pr1 (pr2 precategory-of-elements-presheaf-Precategory) =
-    hom-set-precategory-of-elements-presheaf-Precategory
-  pr1 (pr1 (pr2 (pr2 precategory-of-elements-presheaf-Precategory))) =
-    comp-hom-precategory-of-elements-presheaf-Precategory
-  pr1 (pr2 (pr1 (pr2 (pr2 precategory-of-elements-presheaf-Precategory))) h g f)
-    =
-    associative-comp-hom-precategory-of-elements-presheaf-Precategory h g f
-  pr2 (pr2 (pr1 (pr2 (pr2 precategory-of-elements-presheaf-Precategory))) h g f)
-    =
-    inv-associative-comp-hom-precategory-of-elements-presheaf-Precategory h g f
-  pr1 (pr2 (pr2 (pr2 precategory-of-elements-presheaf-Precategory))) X =
-    id-hom-precategory-of-elements-presheaf-Precategory
-  pr1 (pr2 (pr2 (pr2 (pr2 precategory-of-elements-presheaf-Precategory)))) =
-    left-unit-law-comp-hom-precategory-of-elements-presheaf-Precategory
-  pr2 (pr2 (pr2 (pr2 (pr2 precategory-of-elements-presheaf-Precategory)))) =
-    right-unit-law-comp-hom-precategory-of-elements-presheaf-Precategory
+  precategory-of-elements-presheaf-Precategory =
+    make-Precategory
+      ( obj-precategory-of-elements-presheaf-Precategory)
+      ( hom-set-precategory-of-elements-presheaf-Precategory)
+      ( comp-hom-precategory-of-elements-presheaf-Precategory)
+      ( λ X → id-hom-precategory-of-elements-presheaf-Precategory {X})
+      ( associative-comp-hom-precategory-of-elements-presheaf-Precategory)
+      ( left-unit-law-comp-hom-precategory-of-elements-presheaf-Precategory)
+      ( right-unit-law-comp-hom-precategory-of-elements-presheaf-Precategory)
 ```
 
 ### The projection from the category of elements of a presheaf to the base category
diff --git a/src/category-theory/precategory-of-functors-from-small-to-large-precategories.lagda.md b/src/category-theory/precategory-of-functors-from-small-to-large-precategories.lagda.md
index b1527d41a3..6494609f6b 100644
--- a/src/category-theory/precategory-of-functors-from-small-to-large-precategories.lagda.md
+++ b/src/category-theory/precategory-of-functors-from-small-to-large-precategories.lagda.md
@@ -13,6 +13,7 @@ open import category-theory.natural-transformations-functors-from-small-to-large
 open import category-theory.precategories
 
 open import foundation.identity-types
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -78,7 +79,7 @@ module _
     associative-comp-natural-transformation-Small-Large-Precategory
       C D F G H I f g h
 
-  inv-associative-comp-hom-functor-large-precategory-Small-Large-Precategory :
+  involutive-eq-associative-comp-hom-functor-large-precategory-Small-Large-Precategory :
     {γF γG γH γI : Level}
     {F : functor-Small-Large-Precategory C D γF}
     {G : functor-Small-Large-Precategory C D γG}
@@ -87,15 +88,15 @@ module _
     (h : natural-transformation-Small-Large-Precategory C D H I)
     (g : natural-transformation-Small-Large-Precategory C D G H)
     (f : natural-transformation-Small-Large-Precategory C D F G) →
-    comp-natural-transformation-Small-Large-Precategory C D F H I
-      ( h)
-      ( comp-natural-transformation-Small-Large-Precategory C D F G H g f) ＝
     comp-natural-transformation-Small-Large-Precategory C D F G I
       ( comp-natural-transformation-Small-Large-Precategory C D G H I h g)
-      ( f)
-  inv-associative-comp-hom-functor-large-precategory-Small-Large-Precategory
+      ( f) ＝ⁱ
+    comp-natural-transformation-Small-Large-Precategory C D F H I
+      ( h)
+      ( comp-natural-transformation-Small-Large-Precategory C D F G H g f)
+  involutive-eq-associative-comp-hom-functor-large-precategory-Small-Large-Precategory
     {F = F} {G} {H} {I} h g f =
-    inv-associative-comp-natural-transformation-Small-Large-Precategory
+    involutive-eq-associative-comp-natural-transformation-Small-Large-Precategory
       C D F G H I f g h
 
   id-hom-functor-large-precategory-Small-Large-Precategory :
@@ -140,13 +141,9 @@ module _
   id-hom-Large-Precategory
     functor-large-precategory-Small-Large-Precategory {X = F} =
     id-hom-functor-large-precategory-Small-Large-Precategory {F = F}
-  associative-comp-hom-Large-Precategory
-    functor-large-precategory-Small-Large-Precategory {X = F} {G} {H} {I} =
-    associative-comp-hom-functor-large-precategory-Small-Large-Precategory
-      { F = F} {G} {H} {I}
-  inv-associative-comp-hom-Large-Precategory
+  involutive-eq-associative-comp-hom-Large-Precategory
     functor-large-precategory-Small-Large-Precategory {X = F} {G} {H} {I} =
-    inv-associative-comp-hom-functor-large-precategory-Small-Large-Precategory
+    involutive-eq-associative-comp-hom-functor-large-precategory-Small-Large-Precategory
       { F = F} {G} {H} {I}
   left-unit-law-comp-hom-Large-Precategory
     functor-large-precategory-Small-Large-Precategory {X = F} {G} =
diff --git a/src/category-theory/precategory-of-functors.lagda.md b/src/category-theory/precategory-of-functors.lagda.md
index 3850b59294..9130ba3fe0 100644
--- a/src/category-theory/precategory-of-functors.lagda.md
+++ b/src/category-theory/precategory-of-functors.lagda.md
@@ -21,6 +21,7 @@ open import foundation.function-extensionality
 open import foundation.functoriality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositions
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -67,20 +68,20 @@ module _
     associative-comp-natural-transformation-Precategory
       C D F G H I f g h
 
-  inv-associative-comp-hom-functor-precategory-Precategory :
+  involutive-eq-associative-comp-hom-functor-precategory-Precategory :
     {F G H I : functor-Precategory C D}
     (h : natural-transformation-Precategory C D H I)
     (g : natural-transformation-Precategory C D G H)
     (f : natural-transformation-Precategory C D F G) →
-    comp-natural-transformation-Precategory C D F H I
-      ( h)
-      ( comp-natural-transformation-Precategory C D F G H g f) ＝
     comp-natural-transformation-Precategory C D F G I
       ( comp-natural-transformation-Precategory C D G H I h g)
-      ( f)
-  inv-associative-comp-hom-functor-precategory-Precategory
+      ( f) ＝ⁱ
+    comp-natural-transformation-Precategory C D F H I
+      ( h)
+      ( comp-natural-transformation-Precategory C D F G H g f)
+  involutive-eq-associative-comp-hom-functor-precategory-Precategory
     { F} {G} {H} {I} h g f =
-    inv-associative-comp-natural-transformation-Precategory
+    involutive-eq-associative-comp-natural-transformation-Precategory
       C D F G H I f g h
 
   associative-composition-operation-functor-precategory-Precategory :
@@ -89,16 +90,10 @@ module _
   pr1 associative-composition-operation-functor-precategory-Precategory
     {F} {G} {H} =
     comp-hom-functor-precategory-Precategory {F} {G} {H}
-  pr1
-    ( pr2
-        associative-composition-operation-functor-precategory-Precategory
-          { F} {G} {H} {I} h g f) =
-    associative-comp-hom-functor-precategory-Precategory {F} {G} {H} {I} h g f
   pr2
-    ( pr2
-        associative-composition-operation-functor-precategory-Precategory
-          { F} {G} {H} {I} h g f) =
-    inv-associative-comp-hom-functor-precategory-Precategory
+    associative-composition-operation-functor-precategory-Precategory
+      { F} {G} {H} {I} h g f =
+    involutive-eq-associative-comp-hom-functor-precategory-Precategory
       { F} {G} {H} {I} h g f
 
   id-hom-functor-precategory-Precategory :
diff --git a/src/category-theory/precategory-of-maps-from-small-to-large-precategories.lagda.md b/src/category-theory/precategory-of-maps-from-small-to-large-precategories.lagda.md
index 02cf9d0884..7c755df183 100644
--- a/src/category-theory/precategory-of-maps-from-small-to-large-precategories.lagda.md
+++ b/src/category-theory/precategory-of-maps-from-small-to-large-precategories.lagda.md
@@ -13,6 +13,7 @@ open import category-theory.natural-transformations-maps-from-small-to-large-pre
 open import category-theory.precategories
 
 open import foundation.identity-types
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -70,7 +71,7 @@ module _
     associative-comp-natural-transformation-map-Small-Large-Precategory
       C D F G H I f g h
 
-  inv-associative-comp-hom-map-large-precategory-Small-Large-Precategory :
+  involutive-eq-associative-comp-hom-map-large-precategory-Small-Large-Precategory :
     {γF γG γH γI : Level}
     {F : map-Small-Large-Precategory C D γF}
     {G : map-Small-Large-Precategory C D γG}
@@ -79,15 +80,15 @@ module _
     (h : natural-transformation-map-Small-Large-Precategory C D H I)
     (g : natural-transformation-map-Small-Large-Precategory C D G H)
     (f : natural-transformation-map-Small-Large-Precategory C D F G) →
-    comp-natural-transformation-map-Small-Large-Precategory C D F H I
-      ( h)
-      ( comp-natural-transformation-map-Small-Large-Precategory C D F G H g f) ＝
     comp-natural-transformation-map-Small-Large-Precategory C D F G I
       ( comp-natural-transformation-map-Small-Large-Precategory C D G H I h g)
-      ( f)
-  inv-associative-comp-hom-map-large-precategory-Small-Large-Precategory
+      ( f) ＝ⁱ
+    comp-natural-transformation-map-Small-Large-Precategory C D F H I
+      ( h)
+      ( comp-natural-transformation-map-Small-Large-Precategory C D F G H g f)
+  involutive-eq-associative-comp-hom-map-large-precategory-Small-Large-Precategory
     {F = F} {G} {H} {I} h g f =
-    inv-associative-comp-natural-transformation-map-Small-Large-Precategory
+    involutive-eq-associative-comp-natural-transformation-map-Small-Large-Precategory
       C D F G H I f g h
 
   id-hom-map-large-precategory-Small-Large-Precategory :
@@ -132,12 +133,9 @@ module _
     comp-hom-map-large-precategory-Small-Large-Precategory
   id-hom-Large-Precategory map-large-precategory-Small-Large-Precategory =
     id-hom-map-large-precategory-Small-Large-Precategory
-  associative-comp-hom-Large-Precategory
-    map-large-precategory-Small-Large-Precategory =
-    associative-comp-hom-map-large-precategory-Small-Large-Precategory
-  inv-associative-comp-hom-Large-Precategory
+  involutive-eq-associative-comp-hom-Large-Precategory
     map-large-precategory-Small-Large-Precategory =
-    inv-associative-comp-hom-map-large-precategory-Small-Large-Precategory
+    involutive-eq-associative-comp-hom-map-large-precategory-Small-Large-Precategory
   left-unit-law-comp-hom-Large-Precategory
     map-large-precategory-Small-Large-Precategory =
     left-unit-law-comp-hom-map-large-precategory-Small-Large-Precategory
diff --git a/src/category-theory/precategory-of-maps-precategories.lagda.md b/src/category-theory/precategory-of-maps-precategories.lagda.md
index 5424b56506..bb248ce3b4 100644
--- a/src/category-theory/precategory-of-maps-precategories.lagda.md
+++ b/src/category-theory/precategory-of-maps-precategories.lagda.md
@@ -21,6 +21,7 @@ open import foundation.function-extensionality
 open import foundation.functoriality-dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositions
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -67,19 +68,20 @@ module _
     associative-comp-natural-transformation-map-Precategory
       C D F G H I f g h
 
-  inv-associative-comp-hom-map-precategory-Precategory :
+  involutive-eq-associative-comp-hom-map-precategory-Precategory :
     {F G H I : map-Precategory C D}
     (h : natural-transformation-map-Precategory C D H I)
     (g : natural-transformation-map-Precategory C D G H)
     (f : natural-transformation-map-Precategory C D F G) →
-    comp-natural-transformation-map-Precategory C D F H I
-      ( h)
-      ( comp-natural-transformation-map-Precategory C D F G H g f) ＝
     comp-natural-transformation-map-Precategory C D F G I
       ( comp-natural-transformation-map-Precategory C D G H I h g)
-      ( f)
-  inv-associative-comp-hom-map-precategory-Precategory {F} {G} {H} {I} h g f =
-    inv-associative-comp-natural-transformation-map-Precategory
+      ( f) ＝ⁱ
+    comp-natural-transformation-map-Precategory C D F H I
+      ( h)
+      ( comp-natural-transformation-map-Precategory C D F G H g f)
+  involutive-eq-associative-comp-hom-map-precategory-Precategory
+    { F} {G} {H} {I} h g f =
+    involutive-eq-associative-comp-natural-transformation-map-Precategory
       C D F G H I f g h
 
   associative-composition-operation-map-precategory-Precategory :
@@ -88,16 +90,11 @@ module _
   pr1 associative-composition-operation-map-precategory-Precategory
     {F} {G} {H} =
     comp-hom-map-precategory-Precategory {F} {G} {H}
-  pr1
-    ( pr2
-        associative-composition-operation-map-precategory-Precategory
-          { F} {G} {H} {I} h g f) =
-    associative-comp-hom-map-precategory-Precategory {F} {G} {H} {I} h g f
   pr2
-    ( pr2
-        associative-composition-operation-map-precategory-Precategory
-          { F} {G} {H} {I} h g f) =
-    inv-associative-comp-hom-map-precategory-Precategory {F} {G} {H} {I} h g f
+    associative-composition-operation-map-precategory-Precategory
+      {F} {G} {H} {I} h g f =
+    involutive-eq-associative-comp-hom-map-precategory-Precategory
+      { F} {G} {H} {I} h g f
 
   id-hom-map-precategory-Precategory :
     (F : map-Precategory C D) → natural-transformation-map-Precategory C D F F
diff --git a/src/category-theory/pregroupoids.lagda.md b/src/category-theory/pregroupoids.lagda.md
index baf074016f..e5cd73ad42 100644
--- a/src/category-theory/pregroupoids.lagda.md
+++ b/src/category-theory/pregroupoids.lagda.md
@@ -16,6 +16,7 @@ open import foundation.identity-types
 open import foundation.iterated-dependent-product-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.type-arithmetic-dependent-pair-types
 open import foundation.universe-levels
 ```
@@ -95,13 +96,13 @@ module _
   associative-comp-hom-Pregroupoid =
     associative-comp-hom-Precategory precategory-Pregroupoid
 
-  inv-associative-comp-hom-Pregroupoid :
+  involutive-eq-associative-comp-hom-Pregroupoid :
     {x y z w : obj-Pregroupoid} (h : hom-Pregroupoid z w)
     (g : hom-Pregroupoid y z) (f : hom-Pregroupoid x y) →
-    comp-hom-Pregroupoid h (comp-hom-Pregroupoid g f) ＝
-    comp-hom-Pregroupoid (comp-hom-Pregroupoid h g) f
-  inv-associative-comp-hom-Pregroupoid =
-    inv-associative-comp-hom-Precategory precategory-Pregroupoid
+    comp-hom-Pregroupoid (comp-hom-Pregroupoid h g) f ＝ⁱ
+    comp-hom-Pregroupoid h (comp-hom-Pregroupoid g f)
+  involutive-eq-associative-comp-hom-Pregroupoid =
+    involutive-eq-associative-comp-hom-Precategory precategory-Pregroupoid
 
   left-unit-law-comp-hom-Pregroupoid :
     {x y : obj-Pregroupoid} (f : hom-Pregroupoid x y) →
diff --git a/src/category-theory/presheaf-categories.lagda.md b/src/category-theory/presheaf-categories.lagda.md
index 0327297765..1e61775687 100644
--- a/src/category-theory/presheaf-categories.lagda.md
+++ b/src/category-theory/presheaf-categories.lagda.md
@@ -23,6 +23,7 @@ open import foundation.function-types
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -209,7 +210,7 @@ module _
       { Z}
       { W}
 
-  inv-associative-comp-hom-presheaf-Precategory :
+  involutive-eq-associative-comp-hom-presheaf-Precategory :
     {l3 l4 l5 l6 : Level}
     (X : presheaf-Precategory l3)
     (Y : presheaf-Precategory l4)
@@ -218,16 +219,18 @@ module _
     (h : hom-presheaf-Precategory Z W)
     (g : hom-presheaf-Precategory Y Z)
     (f : hom-presheaf-Precategory X Y) →
-    comp-hom-presheaf-Precategory X Z W
-      ( h)
-      ( comp-hom-presheaf-Precategory X Y Z g f) ＝
     comp-hom-presheaf-Precategory X Y W
       ( comp-hom-presheaf-Precategory Y Z W h g)
-      ( f)
-  inv-associative-comp-hom-presheaf-Precategory X Y Z W =
-    inv-associative-comp-hom-Large-Precategory
-      ( presheaf-large-precategory-Precategory)
-      { X = X} {Y} {Z} {W}
+      ( f) ＝ⁱ
+    comp-hom-presheaf-Precategory X Z W h
+      ( comp-hom-presheaf-Precategory X Y Z g f)
+  involutive-eq-associative-comp-hom-presheaf-Precategory X Y Z W =
+    involutive-eq-associative-comp-hom-Large-Category
+      ( presheaf-large-category-Precategory)
+      { X = X}
+      { Y}
+      { Z}
+      { W}
 
   left-unit-law-comp-hom-presheaf-Precategory :
     {l3 l4 : Level}
diff --git a/src/category-theory/preunivalent-categories.lagda.md b/src/category-theory/preunivalent-categories.lagda.md
index 7ce5890611..9f593f0cbe 100644
--- a/src/category-theory/preunivalent-categories.lagda.md
+++ b/src/category-theory/preunivalent-categories.lagda.md
@@ -18,6 +18,7 @@ open import foundation.embeddings
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 ```
 
@@ -120,15 +121,16 @@ module _
   associative-comp-hom-Preunivalent-Category =
     associative-comp-hom-Precategory precategory-Preunivalent-Category
 
-  inv-associative-comp-hom-Preunivalent-Category :
+  involutive-eq-associative-comp-hom-Preunivalent-Category :
     {x y z w : obj-Preunivalent-Category}
     (h : hom-Preunivalent-Category z w)
     (g : hom-Preunivalent-Category y z)
     (f : hom-Preunivalent-Category x y) →
-    comp-hom-Preunivalent-Category h (comp-hom-Preunivalent-Category g f) ＝
-    comp-hom-Preunivalent-Category (comp-hom-Preunivalent-Category h g) f
-  inv-associative-comp-hom-Preunivalent-Category =
-    inv-associative-comp-hom-Precategory precategory-Preunivalent-Category
+    comp-hom-Preunivalent-Category (comp-hom-Preunivalent-Category h g) f ＝ⁱ
+    comp-hom-Preunivalent-Category h (comp-hom-Preunivalent-Category g f)
+  involutive-eq-associative-comp-hom-Preunivalent-Category =
+    involutive-eq-associative-comp-hom-Precategory
+      ( precategory-Preunivalent-Category)
 
   associative-composition-operation-Preunivalent-Category :
     associative-composition-operation-binary-family-Set
diff --git a/src/category-theory/products-of-precategories.lagda.md b/src/category-theory/products-of-precategories.lagda.md
index ea5d88bdc8..cb52fdd0e6 100644
--- a/src/category-theory/products-of-precategories.lagda.md
+++ b/src/category-theory/products-of-precategories.lagda.md
@@ -12,6 +12,7 @@ open import category-theory.precategories
 open import foundation.cartesian-product-types
 open import foundation.dependent-pair-types
 open import foundation.equality-cartesian-product-types
+open import foundation.identity-types
 open import foundation.sets
 open import foundation.universe-levels
 ```
@@ -20,11 +21,13 @@ open import foundation.universe-levels
 
 ## Idea
 
-The **product** of two [precategories](category-theory.precategories.md) `C` and
-`D` has as objects pairs `(x , y)`, for `x` in `obj-Precategory C` and `y` in
-`obj-Precategory D`; and has a morphism `Hom (x , y) (x' , y)` for each pair of
-morphisms `f : x → x'` and `g : y → y'`. Composition of morphisms is given by
-composing each entry.
+The
+{{#concept "product" Disambiguation="of precategories" Agda=product-Precategory}}
+of two [precategories](category-theory.precategories.md) `C` and `D` has as
+objects [pairs](foundation-core.cartesian-product-types.md) `(x , y)`, where `x`
+is an object of `C` and `y` is an object of `D`, and has as morphisms from
+`(x , y)` to `(x' , y)` pairs `(f , g)` where `f : x → x'` in `C` and
+`g : y → y'` in `D`. Composition of morphisms is given componentwise.
 
 ## Definition
 
@@ -35,35 +38,73 @@ module _
   (D : Precategory l3 l4)
   where
 
-  product-Precategory :
-    Precategory (l1 ⊔ l3) (l2 ⊔ l4)
-  pr1 product-Precategory = obj-Precategory C × obj-Precategory D
-  pr1 (pr2 product-Precategory) (x , y) (x' , y') =
+  obj-product-Precategory : UU (l1 ⊔ l3)
+  obj-product-Precategory = obj-Precategory C × obj-Precategory D
+
+  hom-set-product-Precategory :
+    obj-product-Precategory → obj-product-Precategory → Set (l2 ⊔ l4)
+  hom-set-product-Precategory (x , y) (x' , y') =
     product-Set (hom-set-Precategory C x x') (hom-set-Precategory D y y')
-  pr1 (pr1 (pr1 (pr2 (pr2 product-Precategory))) (f' , g') (f , g)) =
-    comp-hom-Precategory C f' f
-  pr2 (pr1 (pr1 (pr2 (pr2 product-Precategory))) (f' , g') (f , g)) =
-    comp-hom-Precategory D g' g
-  pr1
-    ( pr2 (pr1 (pr2 (pr2 product-Precategory))) (f'' , g'') (f' , g') (f , g)) =
+
+  hom-product-Precategory :
+    obj-product-Precategory → obj-product-Precategory → UU (l2 ⊔ l4)
+  hom-product-Precategory p q = type-Set (hom-set-product-Precategory p q)
+
+  is-set-hom-product-Precategory :
+    (p q : obj-product-Precategory) → is-set (hom-product-Precategory p q)
+  is-set-hom-product-Precategory p q =
+    is-set-type-Set (hom-set-product-Precategory p q)
+
+  comp-hom-product-Precategory :
+    {p q r : obj-product-Precategory}
+    (g : hom-product-Precategory q r)
+    (f : hom-product-Precategory p q) →
+    hom-product-Precategory p r
+  comp-hom-product-Precategory (f' , g') (f , g) =
+    ( comp-hom-Precategory C f' f , comp-hom-Precategory D g' g)
+
+  id-hom-product-Precategory :
+    {p : obj-product-Precategory} → hom-product-Precategory p p
+  id-hom-product-Precategory = id-hom-Precategory C , id-hom-Precategory D
+
+  associative-comp-hom-product-Precategory :
+    {p q r s : obj-product-Precategory}
+    (h : hom-product-Precategory r s)
+    (g : hom-product-Precategory q r)
+    (f : hom-product-Precategory p q) →
+    comp-hom-product-Precategory (comp-hom-product-Precategory h g) f ＝
+    comp-hom-product-Precategory h (comp-hom-product-Precategory g f)
+  associative-comp-hom-product-Precategory (f'' , g'') (f' , g') (f , g) =
     eq-pair
       ( associative-comp-hom-Precategory C f'' f' f)
       ( associative-comp-hom-Precategory D g'' g' g)
-  pr2
-    ( pr2 (pr1 (pr2 (pr2 product-Precategory))) (f'' , g'') (f' , g') (f , g)) =
-    eq-pair
-      ( inv-associative-comp-hom-Precategory C f'' f' f)
-      ( inv-associative-comp-hom-Precategory D g'' g' g)
-  pr1 (pr1 (pr2 (pr2 (pr2 product-Precategory))) (x , y)) =
-    id-hom-Precategory C {x}
-  pr2 (pr1 (pr2 (pr2 (pr2 product-Precategory))) (x , y)) =
-    id-hom-Precategory D {y}
-  pr1 (pr2 (pr2 (pr2 (pr2 product-Precategory)))) (f , g) =
+
+  left-unit-law-comp-hom-product-Precategory :
+    {p q : obj-product-Precategory}
+    (f : hom-product-Precategory p q) →
+    comp-hom-product-Precategory id-hom-product-Precategory f ＝ f
+  left-unit-law-comp-hom-product-Precategory (f , g) =
     eq-pair
       ( left-unit-law-comp-hom-Precategory C f)
       ( left-unit-law-comp-hom-Precategory D g)
-  pr2 (pr2 (pr2 (pr2 (pr2 product-Precategory)))) (f , g) =
+
+  right-unit-law-comp-hom-product-Precategory :
+    {p q : obj-product-Precategory}
+    (f : hom-product-Precategory p q) →
+    comp-hom-product-Precategory f id-hom-product-Precategory ＝ f
+  right-unit-law-comp-hom-product-Precategory (f , g) =
     eq-pair
       ( right-unit-law-comp-hom-Precategory C f)
       ( right-unit-law-comp-hom-Precategory D g)
+
+  product-Precategory : Precategory (l1 ⊔ l3) (l2 ⊔ l4)
+  product-Precategory =
+    make-Precategory
+      ( obj-product-Precategory)
+      ( hom-set-product-Precategory)
+      ( comp-hom-product-Precategory)
+      ( λ x → id-hom-product-Precategory {x})
+      ( associative-comp-hom-product-Precategory)
+      ( left-unit-law-comp-hom-product-Precategory)
+      ( right-unit-law-comp-hom-product-Precategory)
 ```
diff --git a/src/category-theory/pseudomonic-functors-precategories.lagda.md b/src/category-theory/pseudomonic-functors-precategories.lagda.md
index fb87d82f14..8dacf40fbd 100644
--- a/src/category-theory/pseudomonic-functors-precategories.lagda.md
+++ b/src/category-theory/pseudomonic-functors-precategories.lagda.md
@@ -30,11 +30,13 @@ open import foundation.universe-levels
 ## Idea
 
 A [functor](category-theory.functors-precategories.md) between
-[precategories](category-theory.precategories.md) `C` and `D` is **pseudomonic**
+[precategories](category-theory.precategories.md) `C` and `D` is
+{{#concept "pseudomonic" Disambiguation="functor between precategories" Agda=is-pseudomonic-functor-Precategory}}
 if it is [faithful](category-theory.faithful-functors-precategories.md) on all
 morphism-[sets](foundation-core.sets.md) and full on
 [isomorphisms](category-theory.isomorphisms-in-precategories.md). In particular,
-this means it induces an equivalence on isomorphism-sets.
+this means it induces an [equivalence](foundation-core.equivalences.md) on
+isomorphism-sets.
 
 Pseudomonic functors present
 [replete subprecategories](category-theory.replete-subprecategories.md), which
@@ -43,7 +45,7 @@ invariance under equivalences_.
 
 ## Definition
 
-### The predicate of being full on isomorphisms
+### The predicate on isomorphisms of being full
 
 ```agda
 module _
@@ -210,7 +212,7 @@ module _
     map-equiv inv-equiv-iso-is-pseudomonic-functor-Precategory
 ```
 
-The previous entry records what is also known as "essential injectivivty".
+The previous entry records what is also known as "essential injectivity".
 
 ```agda
 module _
@@ -247,7 +249,7 @@ module _
     map-equiv inv-equiv-iso-pseudomonic-functor-Precategory
 ```
 
-The previous entry records what is also known as "essential injectivivty".
+The previous entry records what is also known as "essential injectivity".
 
 ### Pseudomonic functors are conservative
 
@@ -291,7 +293,7 @@ module _
 
 - Pseudomonic functors present
   [replete subprecategories](category-theory.replete-subprecategories.md).
-- [Fully faithful functors between precategories](category-theory.pseudomonic-functors-precategories.md)
+- [Fully faithful functors between precategories](category-theory.fully-faithful-functors-precategories.md)
 
 ## External links
 
diff --git a/src/category-theory/representing-arrow-category.lagda.md b/src/category-theory/representing-arrow-category.lagda.md
index 6b9f51b389..ff5fd7b484 100644
--- a/src/category-theory/representing-arrow-category.lagda.md
+++ b/src/category-theory/representing-arrow-category.lagda.md
@@ -8,7 +8,6 @@ module category-theory.representing-arrow-category where
 
 ```agda
 open import category-theory.categories
-open import category-theory.composition-operations-on-binary-families-of-sets
 open import category-theory.isomorphisms-in-precategories
 open import category-theory.precategories
 
@@ -77,30 +76,6 @@ associative-comp-hom-representing-arrow-Category
   refl
 associative-comp-hom-representing-arrow-Category {false} h g f = refl
 
-inv-associative-comp-hom-representing-arrow-Category :
-  {x y z w : obj-representing-arrow-Category} →
-  (h : hom-representing-arrow-Category z w)
-  (g : hom-representing-arrow-Category y z)
-  (f : hom-representing-arrow-Category x y) →
-  ( comp-hom-representing-arrow-Category
-    { x} h (comp-hom-representing-arrow-Category {x} g f)) ＝
-  ( comp-hom-representing-arrow-Category
-    { x} (comp-hom-representing-arrow-Category {y} h g) f)
-inv-associative-comp-hom-representing-arrow-Category
-  { true} {true} {true} {true} h g f =
-  refl
-inv-associative-comp-hom-representing-arrow-Category {false} h g f = refl
-
-associative-composition-operation-representing-arrow-Category :
-  associative-composition-operation-binary-family-Set
-    ( hom-set-representing-arrow-Category)
-pr1 associative-composition-operation-representing-arrow-Category {x} =
-  comp-hom-representing-arrow-Category {x}
-pr1 (pr2 associative-composition-operation-representing-arrow-Category h g f) =
-  associative-comp-hom-representing-arrow-Category h g f
-pr2 (pr2 associative-composition-operation-representing-arrow-Category h g f) =
-  inv-associative-comp-hom-representing-arrow-Category h g f
-
 id-hom-representing-arrow-Category :
   {x : obj-representing-arrow-Category} → hom-representing-arrow-Category x x
 id-hom-representing-arrow-Category {true} = star
@@ -124,24 +99,17 @@ right-unit-law-comp-hom-representing-arrow-Category :
 right-unit-law-comp-hom-representing-arrow-Category {true} {true} f = refl
 right-unit-law-comp-hom-representing-arrow-Category {false} f = refl
 
-is-unital-composition-operation-representing-arrow-Category :
-  is-unital-composition-operation-binary-family-Set
+representing-arrow-Precategory : Precategory lzero lzero
+representing-arrow-Precategory =
+  make-Precategory
+    ( obj-representing-arrow-Category)
     ( hom-set-representing-arrow-Category)
     ( λ {x} {y} {z} → comp-hom-representing-arrow-Category {x} {y} {z})
-pr1 is-unital-composition-operation-representing-arrow-Category x =
-  id-hom-representing-arrow-Category {x}
-pr1 (pr2 is-unital-composition-operation-representing-arrow-Category) =
-  left-unit-law-comp-hom-representing-arrow-Category
-pr2 (pr2 is-unital-composition-operation-representing-arrow-Category) =
-  right-unit-law-comp-hom-representing-arrow-Category
-
-representing-arrow-Precategory : Precategory lzero lzero
-pr1 representing-arrow-Precategory = obj-representing-arrow-Category
-pr1 (pr2 representing-arrow-Precategory) = hom-set-representing-arrow-Category
-pr1 (pr2 (pr2 representing-arrow-Precategory)) =
-  associative-composition-operation-representing-arrow-Category
-pr2 (pr2 (pr2 representing-arrow-Precategory)) =
-  is-unital-composition-operation-representing-arrow-Category
+    ( λ x → id-hom-representing-arrow-Category {x})
+    ( λ {x} {y} {z} {w} →
+      associative-comp-hom-representing-arrow-Category {x} {y} {z} {w})
+    ( λ {x} {y} → left-unit-law-comp-hom-representing-arrow-Category {x} {y})
+    ( λ {x} {y} → right-unit-law-comp-hom-representing-arrow-Category {x} {y})
 ```
 
 ### The representing arrow category
diff --git a/src/category-theory/simplex-category.lagda.md b/src/category-theory/simplex-category.lagda.md
index f152fd4a2b..aa311fa6b6 100644
--- a/src/category-theory/simplex-category.lagda.md
+++ b/src/category-theory/simplex-category.lagda.md
@@ -16,6 +16,7 @@ open import elementary-number-theory.natural-numbers
 open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 
 open import order-theory.order-preserving-maps-posets
@@ -82,19 +83,19 @@ associative-comp-hom-simplex-Category {n} {m} {r} {s} =
     ( Fin-Poset (succ-ℕ r))
     ( Fin-Poset (succ-ℕ s))
 
-inv-associative-comp-hom-simplex-Category :
+involutive-eq-associative-comp-hom-simplex-Category :
   {n m r s : obj-simplex-Category}
   (h : hom-simplex-Category r s)
   (g : hom-simplex-Category m r)
   (f : hom-simplex-Category n m) →
-  comp-hom-simplex-Category {n} {r} {s}
-    ( h)
-    ( comp-hom-simplex-Category {n} {m} {r} g f) ＝
   comp-hom-simplex-Category {n} {m} {s}
     ( comp-hom-simplex-Category {m} {r} {s} h g)
-    ( f)
-inv-associative-comp-hom-simplex-Category {n} {m} {r} {s} =
-  inv-associative-comp-hom-Poset
+    ( f) ＝ⁱ
+  comp-hom-simplex-Category {n} {r} {s}
+    ( h)
+    ( comp-hom-simplex-Category {n} {m} {r} g f)
+involutive-eq-associative-comp-hom-simplex-Category {n} {m} {r} {s} =
+  involutive-eq-associative-comp-hom-Poset
     ( Fin-Poset (succ-ℕ n))
     ( Fin-Poset (succ-ℕ m))
     ( Fin-Poset (succ-ℕ r))
@@ -105,10 +106,7 @@ associative-composition-operation-simplex-Category :
 pr1 associative-composition-operation-simplex-Category {n} {m} {r} =
   comp-hom-simplex-Category {n} {m} {r}
 pr2 associative-composition-operation-simplex-Category {n} {m} {r} {s} =
-  is-associative-witness-associative-composition-operation-binary-family-Set
-    ( hom-set-simplex-Category)
-    ( comp-hom-simplex-Category)
-    ( associative-comp-hom-simplex-Category)
+  involutive-eq-associative-comp-hom-simplex-Category {n} {m} {r} {s}
 
 id-hom-simplex-Category : (n : obj-simplex-Category) → hom-simplex-Category n n
 id-hom-simplex-Category n = id-hom-Poset (Fin-Poset (succ-ℕ n))
diff --git a/src/category-theory/slice-precategories.lagda.md b/src/category-theory/slice-precategories.lagda.md
index f61992ccad..c971f834ce 100644
--- a/src/category-theory/slice-precategories.lagda.md
+++ b/src/category-theory/slice-precategories.lagda.md
@@ -24,6 +24,7 @@ open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
 open import foundation.universe-levels
 ```
@@ -134,18 +135,15 @@ module _
       ( comp-hom-Slice-Precategory h (comp-hom-Slice-Precategory g f))
       ( associative-comp-hom-Precategory C (pr1 h) (pr1 g) (pr1 f))
 
-  inv-associative-comp-hom-Slice-Precategory :
+  involutive-eq-associative-comp-hom-Slice-Precategory :
     {A1 A2 A3 A4 : obj-Slice-Precategory} →
     (h : hom-Slice-Precategory A3 A4)
     (g : hom-Slice-Precategory A2 A3)
     (f : hom-Slice-Precategory A1 A2) →
-    comp-hom-Slice-Precategory h (comp-hom-Slice-Precategory g f) ＝
-    comp-hom-Slice-Precategory (comp-hom-Slice-Precategory h g) f
-  inv-associative-comp-hom-Slice-Precategory h g f =
-    eq-hom-Slice-Precategory
-      ( comp-hom-Slice-Precategory h (comp-hom-Slice-Precategory g f))
-      ( comp-hom-Slice-Precategory (comp-hom-Slice-Precategory h g) f)
-      ( inv-associative-comp-hom-Precategory C (pr1 h) (pr1 g) (pr1 f))
+    comp-hom-Slice-Precategory (comp-hom-Slice-Precategory h g) f ＝ⁱ
+    comp-hom-Slice-Precategory h (comp-hom-Slice-Precategory g f)
+  involutive-eq-associative-comp-hom-Slice-Precategory h g f =
+    involutive-eq-eq (associative-comp-hom-Slice-Precategory h g f)
 ```
 
 ### The left unit law for composition of morphisms in the slice category
@@ -181,10 +179,8 @@ module _
   pr1 Slice-Precategory = obj-Slice-Precategory
   pr1 (pr2 Slice-Precategory) = hom-set-Slice-Precategory
   pr1 (pr1 (pr2 (pr2 Slice-Precategory))) = comp-hom-Slice-Precategory
-  pr1 (pr2 (pr1 (pr2 (pr2 Slice-Precategory))) h g f) =
-    associative-comp-hom-Slice-Precategory h g f
-  pr2 (pr2 (pr1 (pr2 (pr2 Slice-Precategory))) h g f) =
-    inv-associative-comp-hom-Slice-Precategory h g f
+  pr2 (pr1 (pr2 (pr2 Slice-Precategory))) =
+    involutive-eq-associative-comp-hom-Slice-Precategory
   pr1 (pr2 (pr2 (pr2 Slice-Precategory))) = id-hom-Slice-Precategory
   pr1 (pr2 (pr2 (pr2 (pr2 Slice-Precategory)))) =
     left-unit-law-comp-hom-Slice-Precategory
@@ -415,14 +411,12 @@ module _
     eq-pair-eq-fiber
       ( eq-pair-eq-fiber
         ( eq-type-subtype
-          ( λ _ →
-            is-product-prop-Precategory
+          ( is-product-prop-Precategory
               ( Slice-Precategory C A)
               ( X , f)
               ( Y , g)
-              ( _)
-              ( _)
-              ( _))
+              ( Z , comp-hom-Precategory C f h₁)
+              ( h₁ , refl))
           ( refl)))
 
   is-retraction-map-inv-pullback-product-Slice-Precategory :
@@ -434,7 +428,7 @@ module _
       ( eq-pair-eq-fiber
           ( eq-pair-eq-fiber
               ( eq-type-subtype
-                  (λ _ → is-pullback-prop-Precategory C A X Y f g _ _ _ α)
+                  ( λ _ → is-pullback-prop-Precategory C A X Y f g _ _ _ α)
                   ( refl))))
 
   equiv-pullback-product-Slice-Precategory :
diff --git a/src/category-theory/strict-categories.lagda.md b/src/category-theory/strict-categories.lagda.md
index 1a42c41889..d26990c1f5 100644
--- a/src/category-theory/strict-categories.lagda.md
+++ b/src/category-theory/strict-categories.lagda.md
@@ -28,12 +28,15 @@ open import foundation.universe-levels
 
 ## Idea
 
-A **strict category** is a [precategory](category-theory.precategories.md) for
-which the type of objects form a [set](foundation-core.sets.md). Such categories
-are the set-theoretic analogue to
-[(univalent) categories](category-theory.categories.md), and have the defect
-that strict categorical constructions may generally fail to be invariant under
-equivalences.
+A {{#concept "strict category" Agda=Strict-Category}} is a
+[precategory](category-theory.precategories.md) for which the type of objects
+form a [set](foundation-core.sets.md). Such categories are the set-theoretic
+analogue to [(univalent) categories](category-theory.categories.md), and have
+the disadvantages that strict categorical constructions may generally fail to be
+invariant under equivalences, and that the
+([essentially surjective](category-theory.essentially-surjective-functors-precategories.md)/[fully-faithful](category-theory.fully-faithful-functors-precategories.md))-factorization
+system on [functors](category-theory.functors-precategories.md) requires the
+[axiom of choice](foundation.axiom-of-choice.md).
 
 ## Definitions
 
diff --git a/src/category-theory/subcategories.lagda.md b/src/category-theory/subcategories.lagda.md
index 1f4f0b33e8..cb8cf062cf 100644
--- a/src/category-theory/subcategories.lagda.md
+++ b/src/category-theory/subcategories.lagda.md
@@ -28,6 +28,7 @@ open import foundation.fundamental-theorem-of-identity-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtype-identity-principle
 open import foundation.subtypes
 open import foundation.universe-levels
@@ -342,19 +343,19 @@ module _
   associative-comp-hom-Subcategory =
     associative-comp-hom-Subprecategory (precategory-Category C) P
 
-  inv-associative-comp-hom-Subcategory :
+  involutive-eq-associative-comp-hom-Subcategory :
     {x y z w : obj-Subcategory C P}
     (h : hom-Subcategory C P z w)
     (g : hom-Subcategory C P y z)
     (f : hom-Subcategory C P x y) →
-    comp-hom-Subcategory {x} {z} {w}
-      ( h)
-      ( comp-hom-Subcategory {x} {y} {z} g f) ＝
     comp-hom-Subcategory {x} {y} {w}
       ( comp-hom-Subcategory {y} {z} {w} h g)
-      ( f)
-  inv-associative-comp-hom-Subcategory =
-    inv-associative-comp-hom-Subprecategory (precategory-Category C) P
+      ( f) ＝ⁱ
+    comp-hom-Subcategory {x} {z} {w}
+      ( h)
+      ( comp-hom-Subcategory {x} {y} {z} g f)
+  involutive-eq-associative-comp-hom-Subcategory =
+    involutive-eq-associative-comp-hom-Subprecategory (precategory-Category C) P
 
   left-unit-law-comp-hom-Subcategory :
     {x y : obj-Subcategory C P}
diff --git a/src/category-theory/subprecategories.lagda.md b/src/category-theory/subprecategories.lagda.md
index 6ae069a35b..684e900afc 100644
--- a/src/category-theory/subprecategories.lagda.md
+++ b/src/category-theory/subprecategories.lagda.md
@@ -19,6 +19,7 @@ open import foundation.identity-types
 open import foundation.iterated-dependent-product-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
 open import foundation.universe-levels
 ```
@@ -400,26 +401,19 @@ module _
         ( inclusion-hom-Subprecategory C P y z g)
         ( inclusion-hom-Subprecategory C P x y f))
 
-  inv-associative-comp-hom-Subprecategory :
+  involutive-eq-associative-comp-hom-Subprecategory :
     {x y z w : obj-Subprecategory C P}
     (h : hom-Subprecategory C P z w)
     (g : hom-Subprecategory C P y z)
     (f : hom-Subprecategory C P x y) →
-    ( comp-hom-Subprecategory {x} {z} {w} h
-      ( comp-hom-Subprecategory {x} {y} {z} g f)) ＝
-    ( comp-hom-Subprecategory {x} {y} {w}
-      ( comp-hom-Subprecategory {y} {z} {w} h g) f)
-  inv-associative-comp-hom-Subprecategory {x} {y} {z} {w} h g f =
-    eq-type-subtype
-      ( subtype-hom-Subprecategory C P
-        ( inclusion-obj-Subprecategory C P x)
-        ( inclusion-obj-Subprecategory C P w)
-        ( is-in-obj-inclusion-obj-Subprecategory C P x)
-        ( is-in-obj-inclusion-obj-Subprecategory C P w))
-      ( inv-associative-comp-hom-Precategory C
-        ( inclusion-hom-Subprecategory C P z w h)
-        ( inclusion-hom-Subprecategory C P y z g)
-        ( inclusion-hom-Subprecategory C P x y f))
+    comp-hom-Subprecategory {x} {y} {w}
+      ( comp-hom-Subprecategory {y} {z} {w} h g)
+      ( f) ＝ⁱ
+    comp-hom-Subprecategory {x} {z} {w}
+      ( h)
+      ( comp-hom-Subprecategory {x} {y} {z} g f)
+  involutive-eq-associative-comp-hom-Subprecategory {x} {y} {z} {w} h g f =
+    involutive-eq-eq (associative-comp-hom-Subprecategory h g f)
 
   left-unit-law-comp-hom-Subprecategory :
     {x y : obj-Subprecategory C P}
@@ -453,14 +447,8 @@ module _
     associative-composition-operation-binary-family-Set hom-set-Subprecategory
   pr1 associative-composition-operation-Subprecategory {x} {y} {z} =
     comp-hom-Subprecategory {x} {y} {z}
-  pr1
-    ( pr2
-      associative-composition-operation-Subprecategory {x} {y} {z} {w} h g f) =
-    associative-comp-hom-Subprecategory {x} {y} {z} {w} h g f
-  pr2
-    ( pr2
-      associative-composition-operation-Subprecategory {x} {y} {z} {w} h g f) =
-    inv-associative-comp-hom-Subprecategory {x} {y} {z} {w} h g f
+  pr2 associative-composition-operation-Subprecategory =
+    involutive-eq-associative-comp-hom-Subprecategory
 
   is-unital-composition-operation-Subprecategory :
     is-unital-composition-operation-binary-family-Set
diff --git a/src/category-theory/terminal-category.lagda.md b/src/category-theory/terminal-category.lagda.md
index 2fab623ae4..b7fb6f334c 100644
--- a/src/category-theory/terminal-category.lagda.md
+++ b/src/category-theory/terminal-category.lagda.md
@@ -8,7 +8,6 @@ module category-theory.terminal-category where
 
 ```agda
 open import category-theory.categories
-open import category-theory.composition-operations-on-binary-families-of-sets
 open import category-theory.constant-functors
 open import category-theory.functors-categories
 open import category-theory.functors-precategories
@@ -75,24 +74,6 @@ associative-comp-hom-terminal-Category :
   comp-hom-terminal-Category {x} h (comp-hom-terminal-Category {x} g f)
 associative-comp-hom-terminal-Category h g f = refl
 
-inv-associative-comp-hom-terminal-Category :
-  {x y z w : obj-terminal-Category} →
-  (h : hom-terminal-Category z w)
-  (g : hom-terminal-Category y z)
-  (f : hom-terminal-Category x y) →
-  comp-hom-terminal-Category {x} h (comp-hom-terminal-Category {x} g f) ＝
-  comp-hom-terminal-Category {x} (comp-hom-terminal-Category {y} h g) f
-inv-associative-comp-hom-terminal-Category h g f = refl
-
-associative-composition-operation-terminal-Category :
-  associative-composition-operation-binary-family-Set hom-set-terminal-Category
-pr1 associative-composition-operation-terminal-Category =
-  comp-hom-terminal-Category
-pr1 (pr2 associative-composition-operation-terminal-Category h g f) =
-  associative-comp-hom-terminal-Category h g f
-pr2 (pr2 associative-composition-operation-terminal-Category h g f) =
-  inv-associative-comp-hom-terminal-Category h g f
-
 id-hom-terminal-Category :
   {x : obj-terminal-Category} → hom-terminal-Category x x
 id-hom-terminal-Category = star
@@ -109,24 +90,16 @@ right-unit-law-comp-hom-terminal-Category :
   comp-hom-terminal-Category {x} f (id-hom-terminal-Category {x}) ＝ f
 right-unit-law-comp-hom-terminal-Category f = refl
 
-is-unital-composition-operation-terminal-Category :
-  is-unital-composition-operation-binary-family-Set
-    ( hom-set-terminal-Category)
-    ( λ {x} {y} {z} → comp-hom-terminal-Category {x} {y} {z})
-pr1 is-unital-composition-operation-terminal-Category _ =
-  id-hom-terminal-Category
-pr1 (pr2 is-unital-composition-operation-terminal-Category) =
-  left-unit-law-comp-hom-terminal-Category
-pr2 (pr2 is-unital-composition-operation-terminal-Category) =
-  right-unit-law-comp-hom-terminal-Category
-
 terminal-Precategory : Precategory lzero lzero
-pr1 terminal-Precategory = obj-terminal-Category
-pr1 (pr2 terminal-Precategory) = hom-set-terminal-Category
-pr1 (pr2 (pr2 terminal-Precategory)) =
-  associative-composition-operation-terminal-Category
-pr2 (pr2 (pr2 terminal-Precategory)) =
-  is-unital-composition-operation-terminal-Category
+terminal-Precategory =
+  make-Precategory
+    ( obj-terminal-Category)
+    ( hom-set-terminal-Category)
+    ( comp-hom-terminal-Category)
+    ( λ x → id-hom-terminal-Category {x})
+    ( associative-comp-hom-terminal-Category)
+    ( left-unit-law-comp-hom-terminal-Category)
+    ( right-unit-law-comp-hom-terminal-Category)
 ```
 
 ### The terminal category
diff --git a/src/category-theory/wide-subcategories.lagda.md b/src/category-theory/wide-subcategories.lagda.md
index 1d82241ead..8b3b4631fd 100644
--- a/src/category-theory/wide-subcategories.lagda.md
+++ b/src/category-theory/wide-subcategories.lagda.md
@@ -28,6 +28,7 @@ open import foundation.identity-types
 open import foundation.iterated-dependent-product-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
 open import foundation.unit-type
 open import foundation.universe-levels
@@ -354,15 +355,17 @@ module _
   associative-comp-hom-Wide-Subcategory =
     associative-comp-hom-Wide-Subprecategory (precategory-Category C) P
 
-  inv-associative-comp-hom-Wide-Subcategory :
+  involutive-eq-associative-comp-hom-Wide-Subcategory :
     {x y z w : obj-Wide-Subcategory C P}
     (h : hom-Wide-Subcategory C P z w)
     (g : hom-Wide-Subcategory C P y z)
     (f : hom-Wide-Subcategory C P x y) →
-    comp-hom-Wide-Subcategory h (comp-hom-Wide-Subcategory g f) ＝
-    comp-hom-Wide-Subcategory (comp-hom-Wide-Subcategory h g) f
-  inv-associative-comp-hom-Wide-Subcategory =
-    inv-associative-comp-hom-Wide-Subprecategory (precategory-Category C) P
+    comp-hom-Wide-Subcategory (comp-hom-Wide-Subcategory h g) f ＝ⁱ
+    comp-hom-Wide-Subcategory h (comp-hom-Wide-Subcategory g f)
+  involutive-eq-associative-comp-hom-Wide-Subcategory =
+    involutive-eq-associative-comp-hom-Wide-Subprecategory
+      ( precategory-Category C)
+      ( P)
 
   left-unit-law-comp-hom-Wide-Subcategory :
     {x y : obj-Wide-Subcategory C P}
diff --git a/src/category-theory/wide-subprecategories.lagda.md b/src/category-theory/wide-subprecategories.lagda.md
index d86fb934d6..598da33856 100644
--- a/src/category-theory/wide-subprecategories.lagda.md
+++ b/src/category-theory/wide-subprecategories.lagda.md
@@ -21,6 +21,7 @@ open import foundation.identity-types
 open import foundation.iterated-dependent-product-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
 open import foundation.unit-type
 open import foundation.universe-levels
@@ -340,23 +341,15 @@ module _
         ( inclusion-hom-Wide-Subprecategory C P y z g)
         ( inclusion-hom-Wide-Subprecategory C P x y f))
 
-  inv-associative-comp-hom-Wide-Subprecategory :
+  involutive-eq-associative-comp-hom-Wide-Subprecategory :
     {x y z w : obj-Wide-Subprecategory C P}
     (h : hom-Wide-Subprecategory C P z w)
     (g : hom-Wide-Subprecategory C P y z)
     (f : hom-Wide-Subprecategory C P x y) →
-    ( comp-hom-Wide-Subprecategory h
-      ( comp-hom-Wide-Subprecategory g f)) ＝
-    ( comp-hom-Wide-Subprecategory
-      ( comp-hom-Wide-Subprecategory h g) f)
-
-  inv-associative-comp-hom-Wide-Subprecategory {x} {y} {z} {w} h g f =
-    eq-type-subtype
-      ( subtype-hom-Wide-Subprecategory C P x w)
-      ( inv-associative-comp-hom-Precategory C
-        ( inclusion-hom-Wide-Subprecategory C P z w h)
-        ( inclusion-hom-Wide-Subprecategory C P y z g)
-        ( inclusion-hom-Wide-Subprecategory C P x y f))
+    comp-hom-Wide-Subprecategory (comp-hom-Wide-Subprecategory h g) f ＝ⁱ
+    comp-hom-Wide-Subprecategory h (comp-hom-Wide-Subprecategory g f)
+  involutive-eq-associative-comp-hom-Wide-Subprecategory h g f =
+    involutive-eq-eq (associative-comp-hom-Wide-Subprecategory h g f)
 
   left-unit-law-comp-hom-Wide-Subprecategory :
     {x y : obj-Wide-Subprecategory C P}
@@ -383,10 +376,8 @@ module _
       ( hom-set-Wide-Subprecategory)
   pr1 associative-composition-operation-Wide-Subprecategory =
     comp-hom-Wide-Subprecategory
-  pr1 (pr2 associative-composition-operation-Wide-Subprecategory h g f) =
-    associative-comp-hom-Wide-Subprecategory h g f
-  pr2 (pr2 associative-composition-operation-Wide-Subprecategory h g f) =
-    inv-associative-comp-hom-Wide-Subprecategory h g f
+  pr2 associative-composition-operation-Wide-Subprecategory =
+    involutive-eq-associative-comp-hom-Wide-Subprecategory
 
   is-unital-composition-operation-Wide-Subprecategory :
     is-unital-composition-operation-binary-family-Set
diff --git a/src/commutative-algebra/category-of-commutative-rings.lagda.md b/src/commutative-algebra/category-of-commutative-rings.lagda.md
index 530c71516b..bd49bfac40 100644
--- a/src/commutative-algebra/category-of-commutative-rings.lagda.md
+++ b/src/commutative-algebra/category-of-commutative-rings.lagda.md
@@ -36,7 +36,7 @@ is-large-category-Commutative-Ring-Large-Category :
 is-large-category-Commutative-Ring-Large-Category =
   is-equiv-iso-eq-Commutative-Ring
 
-Commutative-Ring-Large-Category : Large-Category lsuc _⊔_
+Commutative-Ring-Large-Category : Large-Category lsuc (_⊔_)
 large-precategory-Large-Category Commutative-Ring-Large-Category =
   Commutative-Ring-Large-Precategory
 is-large-category-Large-Category Commutative-Ring-Large-Category =
diff --git a/src/commutative-algebra/commutative-rings.lagda.md b/src/commutative-algebra/commutative-rings.lagda.md
index 1ed5a691d1..06b3e5d600 100644
--- a/src/commutative-algebra/commutative-rings.lagda.md
+++ b/src/commutative-algebra/commutative-rings.lagda.md
@@ -53,23 +53,32 @@ commutative, i.e., if `xy = yx` for all `x, y ∈ A`.
 
 ## Definition
 
-### Commutative rings
+### The predicate on rings of being commutative
 
 ```agda
-is-commutative-Ring :
-  { l : Level} → Ring l → UU l
-is-commutative-Ring A =
-  (x y : type-Ring A) → mul-Ring A x y ＝ mul-Ring A y x
-
-is-prop-is-commutative-Ring :
-  { l : Level} (A : Ring l) → is-prop (is-commutative-Ring A)
-is-prop-is-commutative-Ring A =
-  is-prop-Π
-    ( λ x →
-      is-prop-Π
-      ( λ y →
-        is-set-type-Ring A (mul-Ring A x y) (mul-Ring A y x)))
+module _
+  {l : Level} (A : Ring l)
+  where
+
+  is-commutative-Ring : UU l
+  is-commutative-Ring =
+    (x y : type-Ring A) → mul-Ring A x y ＝ mul-Ring A y x
+
+  is-prop-is-commutative-Ring : is-prop is-commutative-Ring
+  is-prop-is-commutative-Ring =
+    is-prop-Π
+      ( λ x →
+        is-prop-Π
+        ( λ y →
+          is-set-type-Ring A (mul-Ring A x y) (mul-Ring A y x)))
 
+  is-commutative-prop-Ring : Prop l
+  is-commutative-prop-Ring = is-commutative-Ring , is-prop-is-commutative-Ring
+```
+
+### Commutative rings
+
+```agda
 Commutative-Ring :
   ( l : Level) → UU (lsuc l)
 Commutative-Ring l = Σ (Ring l) is-commutative-Ring
@@ -127,7 +136,7 @@ module _
   additive-semigroup-Commutative-Ring = semigroup-Ab ab-Commutative-Ring
 
   is-group-additive-semigroup-Commutative-Ring :
-    is-group additive-semigroup-Commutative-Ring
+    is-group-Semigroup additive-semigroup-Commutative-Ring
   is-group-additive-semigroup-Commutative-Ring =
     is-group-Ab ab-Commutative-Ring
 
@@ -278,7 +287,9 @@ module _
 
 ```agda
   has-negatives-Commutative-Ring :
-    is-group' additive-semigroup-Commutative-Ring has-zero-Commutative-Ring
+    is-group-is-unital-Semigroup
+      ( additive-semigroup-Commutative-Ring)
+      ( has-zero-Commutative-Ring)
   has-negatives-Commutative-Ring = has-negatives-Ab ab-Commutative-Ring
 
   neg-Commutative-Ring : type-Commutative-Ring → type-Commutative-Ring
diff --git a/src/commutative-algebra/commutative-semirings.lagda.md b/src/commutative-algebra/commutative-semirings.lagda.md
index 012521dbb4..78e21f5b30 100644
--- a/src/commutative-algebra/commutative-semirings.lagda.md
+++ b/src/commutative-algebra/commutative-semirings.lagda.md
@@ -12,6 +12,7 @@ open import elementary-number-theory.natural-numbers
 
 open import foundation.dependent-pair-types
 open import foundation.identity-types
+open import foundation.iterated-dependent-product-types
 open import foundation.propositions
 open import foundation.sets
 open import foundation.universe-levels
@@ -26,27 +27,38 @@ open import ring-theory.semirings
 
 ## Idea
 
-A semiring `A` is said to be **commutative** if its multiplicative operation is
-commutative, i.e., if `xy = yx` for all `x, y ∈ A`.
+A [semiring](ring-theory.semirings.md) `A` is said to be
+{{#concept "commutative" Disambiguation="semiring" Agda=is-commutative-Semiring}}
+if its multiplicative operation is commutative, i.e., if `xy = yx` for all
+`x, y ∈ A`.
 
-## Definition
+## Definitions
 
-### Main definition
+### The predicate on semirings of being commutative
 
 ```agda
-is-commutative-Semiring : {l : Level} → Semiring l → UU l
-is-commutative-Semiring A =
-  (x y : type-Semiring A) → mul-Semiring A x y ＝ mul-Semiring A y x
-
-is-prop-is-commutative-Semiring :
-  { l : Level} (A : Semiring l) → is-prop (is-commutative-Semiring A)
-is-prop-is-commutative-Semiring A =
-  is-prop-Π
-    ( λ x →
-      is-prop-Π
-      ( λ y →
-        is-set-type-Semiring A (mul-Semiring A x y) (mul-Semiring A y x)))
+module _
+  {l : Level} (A : Semiring l)
+  where
+
+  is-commutative-Semiring : UU l
+  is-commutative-Semiring =
+    (x y : type-Semiring A) → mul-Semiring A x y ＝ mul-Semiring A y x
+
+  is-prop-is-commutative-Semiring : is-prop is-commutative-Semiring
+  is-prop-is-commutative-Semiring =
+    is-prop-iterated-Π 2
+      ( λ x y →
+        is-set-type-Semiring A (mul-Semiring A x y) (mul-Semiring A y x))
 
+  is-commutative-prop-Semiring : Prop l
+  is-commutative-prop-Semiring =
+    ( is-commutative-Semiring , is-prop-is-commutative-Semiring)
+```
+
+### The type of commutative semirings
+
+```agda
 Commutative-Semiring :
   ( l : Level) → UU (lsuc l)
 Commutative-Semiring l = Σ (Semiring l) is-commutative-Semiring
diff --git a/src/commutative-algebra/euclidean-domains.lagda.md b/src/commutative-algebra/euclidean-domains.lagda.md
index 46a30f31a1..abd586d5ef 100644
--- a/src/commutative-algebra/euclidean-domains.lagda.md
+++ b/src/commutative-algebra/euclidean-domains.lagda.md
@@ -172,7 +172,7 @@ module _
   additive-semigroup-Euclidean-Domain = semigroup-Ab ab-Euclidean-Domain
 
   is-group-additive-semigroup-Euclidean-Domain :
-    is-group additive-semigroup-Euclidean-Domain
+    is-group-Semigroup additive-semigroup-Euclidean-Domain
   is-group-additive-semigroup-Euclidean-Domain =
     is-group-Ab ab-Euclidean-Domain
 
@@ -282,7 +282,9 @@ module _
 
 ```agda
   has-negatives-Euclidean-Domain :
-    is-group' additive-semigroup-Euclidean-Domain has-zero-Euclidean-Domain
+    is-group-is-unital-Semigroup
+      ( additive-semigroup-Euclidean-Domain)
+      ( has-zero-Euclidean-Domain)
   has-negatives-Euclidean-Domain = has-negatives-Ab ab-Euclidean-Domain
 
   neg-Euclidean-Domain : type-Euclidean-Domain → type-Euclidean-Domain
diff --git a/src/commutative-algebra/groups-of-units-commutative-rings.lagda.md b/src/commutative-algebra/groups-of-units-commutative-rings.lagda.md
index 7a04dcd509..f3fcd5aac1 100644
--- a/src/commutative-algebra/groups-of-units-commutative-rings.lagda.md
+++ b/src/commutative-algebra/groups-of-units-commutative-rings.lagda.md
@@ -151,14 +151,14 @@ module _
       ( commutative-mul-Commutative-Ring A (pr1 x) (pr1 y))
 
   is-group-group-of-units-Commutative-Ring' :
-    is-group'
+    is-group-is-unital-Semigroup
       ( semigroup-group-of-units-Commutative-Ring)
       ( is-unital-group-of-units-Commutative-Ring)
   is-group-group-of-units-Commutative-Ring' =
     is-group-group-of-units-Ring' (ring-Commutative-Ring A)
 
   is-group-group-of-units-Commutative-Ring :
-    is-group semigroup-group-of-units-Commutative-Ring
+    is-group-Semigroup semigroup-group-of-units-Commutative-Ring
   is-group-group-of-units-Commutative-Ring =
     is-group-group-of-units-Ring (ring-Commutative-Ring A)
 
diff --git a/src/commutative-algebra/homomorphisms-commutative-rings.lagda.md b/src/commutative-algebra/homomorphisms-commutative-rings.lagda.md
index f2a6ee1bd5..284d752ce5 100644
--- a/src/commutative-algebra/homomorphisms-commutative-rings.lagda.md
+++ b/src/commutative-algebra/homomorphisms-commutative-rings.lagda.md
@@ -397,19 +397,6 @@ module _
       ( h)
       ( g)
       ( f)
-
-  inv-associative-comp-hom-Commutative-Ring :
-    comp-hom-Commutative-Ring A C D h (comp-hom-Commutative-Ring A B C g f) ＝
-    comp-hom-Commutative-Ring A B D (comp-hom-Commutative-Ring B C D h g) f
-  inv-associative-comp-hom-Commutative-Ring =
-    inv-associative-comp-hom-Ring
-      ( ring-Commutative-Ring A)
-      ( ring-Commutative-Ring B)
-      ( ring-Commutative-Ring C)
-      ( ring-Commutative-Ring D)
-      ( h)
-      ( g)
-      ( f)
 ```
 
 ### Unit laws for composition of homomorphisms of commutative rings
diff --git a/src/commutative-algebra/homomorphisms-commutative-semirings.lagda.md b/src/commutative-algebra/homomorphisms-commutative-semirings.lagda.md
index d190260a96..c5ee3254af 100644
--- a/src/commutative-algebra/homomorphisms-commutative-semirings.lagda.md
+++ b/src/commutative-algebra/homomorphisms-commutative-semirings.lagda.md
@@ -361,23 +361,6 @@ module _
       ( h)
       ( g)
       ( f)
-
-  inv-associative-comp-hom-Commutative-Semiring :
-    comp-hom-Commutative-Semiring A C D
-      ( h)
-      ( comp-hom-Commutative-Semiring A B C g f) ＝
-    comp-hom-Commutative-Semiring A B D
-      ( comp-hom-Commutative-Semiring B C D h g)
-      ( f)
-  inv-associative-comp-hom-Commutative-Semiring =
-    inv-associative-comp-hom-Semiring
-      ( semiring-Commutative-Semiring A)
-      ( semiring-Commutative-Semiring B)
-      ( semiring-Commutative-Semiring C)
-      ( semiring-Commutative-Semiring D)
-      ( h)
-      ( g)
-      ( f)
 ```
 
 ### Unit laws for composition of homomorphisms of commutative semirings
diff --git a/src/commutative-algebra/integral-domains.lagda.md b/src/commutative-algebra/integral-domains.lagda.md
index f345c4d755..d960d94053 100644
--- a/src/commutative-algebra/integral-domains.lagda.md
+++ b/src/commutative-algebra/integral-domains.lagda.md
@@ -141,7 +141,7 @@ module _
   additive-semigroup-Integral-Domain = semigroup-Ab ab-Integral-Domain
 
   is-group-additive-semigroup-Integral-Domain :
-    is-group additive-semigroup-Integral-Domain
+    is-group-Semigroup additive-semigroup-Integral-Domain
   is-group-additive-semigroup-Integral-Domain =
     is-group-Ab ab-Integral-Domain
 
@@ -251,7 +251,9 @@ module _
 
 ```agda
   has-negatives-Integral-Domain :
-    is-group' additive-semigroup-Integral-Domain has-zero-Integral-Domain
+    is-group-is-unital-Semigroup
+      ( additive-semigroup-Integral-Domain)
+      ( has-zero-Integral-Domain)
   has-negatives-Integral-Domain = has-negatives-Ab ab-Integral-Domain
 
   neg-Integral-Domain : type-Integral-Domain → type-Integral-Domain
diff --git a/src/commutative-algebra/precategory-of-commutative-rings.lagda.md b/src/commutative-algebra/precategory-of-commutative-rings.lagda.md
index 21baa0f9d1..51c54f1c3f 100644
--- a/src/commutative-algebra/precategory-of-commutative-rings.lagda.md
+++ b/src/commutative-algebra/precategory-of-commutative-rings.lagda.md
@@ -7,52 +7,44 @@ module commutative-algebra.precategory-of-commutative-rings where
 <details><summary>Imports</summary>
 
 ```agda
+open import category-theory.full-large-subprecategories
 open import category-theory.large-precategories
 open import category-theory.precategories
 
 open import commutative-algebra.commutative-rings
-open import commutative-algebra.homomorphisms-commutative-rings
 
 open import foundation.universe-levels
+
+open import ring-theory.precategory-of-rings
 ```
 
 </details>
 
 ## Idea
 
-The **precategory of commutative rings** consists of commutative rings and
-homomorphisms of commutative rings.
+The
+{{#concept "precategory of commutative rings" Agda=Commutative-Ring-Large-Precategory}}
+consists of [commutative rings](commutative-algebra.commutative-rings.md) and
+[homomorphisms of commutative rings](commutative-algebra.homomorphisms-commutative-rings.md).
 
 ## Definitions
 
+### The precategory of commutative rings as a full subprecategory of rings
+
+```agda
+Commutative-Ring-Full-Large-Subprecategory :
+  Full-Large-Subprecategory (λ l → l) Ring-Large-Precategory
+Commutative-Ring-Full-Large-Subprecategory = is-commutative-prop-Ring
+```
+
 ### The large precategory of commutative rings
 
 ```agda
-Commutative-Ring-Large-Precategory : Large-Precategory lsuc _⊔_
-obj-Large-Precategory
-  Commutative-Ring-Large-Precategory =
-  Commutative-Ring
-hom-set-Large-Precategory
-  Commutative-Ring-Large-Precategory =
-  hom-set-Commutative-Ring
-comp-hom-Large-Precategory
-  Commutative-Ring-Large-Precategory {X = A} {B} {C} =
-  comp-hom-Commutative-Ring A B C
-id-hom-Large-Precategory
-  Commutative-Ring-Large-Precategory {X = A} =
-  id-hom-Commutative-Ring A
-associative-comp-hom-Large-Precategory
-  Commutative-Ring-Large-Precategory {X = A} {B} {C} {D} =
-  associative-comp-hom-Commutative-Ring A B C D
-inv-associative-comp-hom-Large-Precategory
-  Commutative-Ring-Large-Precategory {X = A} {B} {C} {D} =
-  inv-associative-comp-hom-Commutative-Ring A B C D
-left-unit-law-comp-hom-Large-Precategory
-  Commutative-Ring-Large-Precategory {X = A} {B} =
-  left-unit-law-comp-hom-Commutative-Ring A B
-right-unit-law-comp-hom-Large-Precategory
-  Commutative-Ring-Large-Precategory {X = A} {B} =
-  right-unit-law-comp-hom-Commutative-Ring A B
+Commutative-Ring-Large-Precategory : Large-Precategory lsuc (_⊔_)
+Commutative-Ring-Large-Precategory =
+  large-precategory-Full-Large-Subprecategory
+    ( Ring-Large-Precategory)
+    ( Commutative-Ring-Full-Large-Subprecategory)
 ```
 
 ### The precategory of commutative rings of universe level `l`
diff --git a/src/commutative-algebra/precategory-of-commutative-semirings.lagda.md b/src/commutative-algebra/precategory-of-commutative-semirings.lagda.md
index ac5b884aee..e0239a074f 100644
--- a/src/commutative-algebra/precategory-of-commutative-semirings.lagda.md
+++ b/src/commutative-algebra/precategory-of-commutative-semirings.lagda.md
@@ -7,52 +7,45 @@ module commutative-algebra.precategory-of-commutative-semirings where
 <details><summary>Imports</summary>
 
 ```agda
+open import category-theory.full-large-subprecategories
 open import category-theory.large-precategories
 open import category-theory.precategories
 
 open import commutative-algebra.commutative-semirings
-open import commutative-algebra.homomorphisms-commutative-semirings
 
 open import foundation.universe-levels
+
+open import ring-theory.precategory-of-semirings
 ```
 
 </details>
 
 ## Idea
 
-The **precategory of commutative semirings** consists of commutative semirings
-and homomorphisms of semirings.
+The
+{{#concept "precategory of commutative semirings" Agda=Commutative-Semiring-Large-Precategory}}
+consists of
+[commutative semirings](commutative-algebra.commutative-semirings.md) and
+[homomorphisms of semirings](commutative-algebra.homomorphisms-commutative-semirings.md).
 
 ## Definitions
 
+### The precategory of commutative semirings as a full subprecategory of semirings
+
+```agda
+Commutative-Semiring-Full-Large-Precategory :
+  Full-Large-Subprecategory (λ l → l) Semiring-Large-Precategory
+Commutative-Semiring-Full-Large-Precategory = is-commutative-prop-Semiring
+```
+
 ### The large precategory of commutative semirings
 
 ```agda
-Commutative-Semiring-Large-Precategory : Large-Precategory lsuc _⊔_
-obj-Large-Precategory
-  Commutative-Semiring-Large-Precategory =
-  Commutative-Semiring
-hom-set-Large-Precategory
-  Commutative-Semiring-Large-Precategory =
-  hom-set-Commutative-Semiring
-comp-hom-Large-Precategory
-  Commutative-Semiring-Large-Precategory {X = A} {B} {C} =
-  comp-hom-Commutative-Semiring A B C
-id-hom-Large-Precategory
-  Commutative-Semiring-Large-Precategory {X = A} =
-  id-hom-Commutative-Semiring A
-associative-comp-hom-Large-Precategory
-  Commutative-Semiring-Large-Precategory {X = A} {B} {C} {D} =
-  associative-comp-hom-Commutative-Semiring A B C D
-inv-associative-comp-hom-Large-Precategory
-  Commutative-Semiring-Large-Precategory {X = A} {B} {C} {D} =
-  inv-associative-comp-hom-Commutative-Semiring A B C D
-left-unit-law-comp-hom-Large-Precategory
-  Commutative-Semiring-Large-Precategory {X = A} {B} =
-  left-unit-law-comp-hom-Commutative-Semiring A B
-right-unit-law-comp-hom-Large-Precategory
-  Commutative-Semiring-Large-Precategory {X = A} {B} =
-  right-unit-law-comp-hom-Commutative-Semiring A B
+Commutative-Semiring-Large-Precategory : Large-Precategory lsuc (_⊔_)
+Commutative-Semiring-Large-Precategory =
+  large-precategory-Full-Large-Subprecategory
+    ( Semiring-Large-Precategory)
+    ( Commutative-Semiring-Full-Large-Precategory)
 ```
 
 ### The precategory of commutative semirings of universe level `l`
diff --git a/src/finite-algebra/commutative-finite-rings.lagda.md b/src/finite-algebra/commutative-finite-rings.lagda.md
index 52394614ad..43b9f67381 100644
--- a/src/finite-algebra/commutative-finite-rings.lagda.md
+++ b/src/finite-algebra/commutative-finite-rings.lagda.md
@@ -147,7 +147,7 @@ module _
   additive-semigroup-Commutative-Ring-𝔽 = semigroup-Ab ab-Commutative-Ring-𝔽
 
   is-group-additive-semigroup-Commutative-Ring-𝔽 :
-    is-group additive-semigroup-Commutative-Ring-𝔽
+    is-group-Semigroup additive-semigroup-Commutative-Ring-𝔽
   is-group-additive-semigroup-Commutative-Ring-𝔽 =
     is-group-Ab ab-Commutative-Ring-𝔽
 
@@ -258,7 +258,9 @@ module _
 
 ```agda
   has-negatives-Commutative-Ring-𝔽 :
-    is-group' additive-semigroup-Commutative-Ring-𝔽 has-zero-Commutative-Ring-𝔽
+    is-group-is-unital-Semigroup
+      ( additive-semigroup-Commutative-Ring-𝔽)
+      ( has-zero-Commutative-Ring-𝔽)
   has-negatives-Commutative-Ring-𝔽 = has-negatives-Ab ab-Commutative-Ring-𝔽
 
   neg-Commutative-Ring-𝔽 : type-Commutative-Ring-𝔽 → type-Commutative-Ring-𝔽
diff --git a/src/finite-algebra/finite-fields.lagda.md b/src/finite-algebra/finite-fields.lagda.md
index f4fbb7921e..829d28547f 100644
--- a/src/finite-algebra/finite-fields.lagda.md
+++ b/src/finite-algebra/finite-fields.lagda.md
@@ -126,7 +126,7 @@ module _
   additive-semigroup-Field-𝔽 = semigroup-Ab ab-Field-𝔽
 
   is-group-additive-semigroup-Field-𝔽 :
-    is-group additive-semigroup-Field-𝔽
+    is-group-Semigroup additive-semigroup-Field-𝔽
   is-group-additive-semigroup-Field-𝔽 =
     is-group-Ab ab-Field-𝔽
 
@@ -230,7 +230,7 @@ module _
 
 ```agda
   has-negatives-Field-𝔽 :
-    is-group' additive-semigroup-Field-𝔽 has-zero-Field-𝔽
+    is-group-is-unital-Semigroup additive-semigroup-Field-𝔽 has-zero-Field-𝔽
   has-negatives-Field-𝔽 = has-negatives-Ab ab-Field-𝔽
 
   neg-Field-𝔽 : type-Field-𝔽 → type-Field-𝔽
diff --git a/src/finite-algebra/finite-rings.lagda.md b/src/finite-algebra/finite-rings.lagda.md
index 0f4df958a6..8d5ca8a2af 100644
--- a/src/finite-algebra/finite-rings.lagda.md
+++ b/src/finite-algebra/finite-rings.lagda.md
@@ -140,7 +140,8 @@ module _
     Id (add-Ring-𝔽 (add-Ring-𝔽 x y) z) (add-Ring-𝔽 x (add-Ring-𝔽 y z))
   associative-add-Ring-𝔽 = associative-add-Ring (ring-Ring-𝔽 R)
 
-  is-group-additive-semigroup-Ring-𝔽 : is-group (additive-semigroup-Ring-𝔽 R)
+  is-group-additive-semigroup-Ring-𝔽 :
+    is-group-Semigroup (additive-semigroup-Ring-𝔽 R)
   is-group-additive-semigroup-Ring-𝔽 =
     is-group-additive-semigroup-Ring (ring-Ring-𝔽 R)
 
@@ -232,7 +233,9 @@ module _
   where
 
   has-negatives-Ring-𝔽 :
-    is-group' (additive-semigroup-Ring-𝔽 R) (has-zero-Ring-𝔽 R)
+    is-group-is-unital-Semigroup
+      ( additive-semigroup-Ring-𝔽 R)
+      ( has-zero-Ring-𝔽 R)
   has-negatives-Ring-𝔽 = has-negatives-Ring (ring-Ring-𝔽 R)
 
   neg-Ring-𝔽 : type-Ring-𝔽 R → type-Ring-𝔽 R
diff --git a/src/finite-algebra/homomorphisms-commutative-finite-rings.lagda.md b/src/finite-algebra/homomorphisms-commutative-finite-rings.lagda.md
index baf6d0942e..bbd41dbe64 100644
--- a/src/finite-algebra/homomorphisms-commutative-finite-rings.lagda.md
+++ b/src/finite-algebra/homomorphisms-commutative-finite-rings.lagda.md
@@ -418,23 +418,6 @@ module _
       ( h)
       ( g)
       ( f)
-
-  inv-associative-comp-hom-Commutative-Ring-𝔽 :
-    comp-hom-Commutative-Ring-𝔽 A C D
-      ( h)
-      ( comp-hom-Commutative-Ring-𝔽 A B C g f) ＝
-    comp-hom-Commutative-Ring-𝔽 A B D
-      ( comp-hom-Commutative-Ring-𝔽 B C D h g)
-      ( f)
-  inv-associative-comp-hom-Commutative-Ring-𝔽 =
-    inv-associative-comp-hom-Ring
-      ( ring-Commutative-Ring-𝔽 A)
-      ( ring-Commutative-Ring-𝔽 B)
-      ( ring-Commutative-Ring-𝔽 C)
-      ( ring-Commutative-Ring-𝔽 D)
-      ( h)
-      ( g)
-      ( f)
 ```
 
 ### Unit laws for composition of homomorphisms of commutative rings
diff --git a/src/finite-algebra/homomorphisms-finite-rings.lagda.md b/src/finite-algebra/homomorphisms-finite-rings.lagda.md
index c469d3fe0f..c533ed1b6c 100644
--- a/src/finite-algebra/homomorphisms-finite-rings.lagda.md
+++ b/src/finite-algebra/homomorphisms-finite-rings.lagda.md
@@ -382,19 +382,6 @@ module _
       ( h)
       ( g)
       ( f)
-
-  inv-associative-comp-hom-Ring-𝔽 :
-    comp-hom-Ring-𝔽 A C D h (comp-hom-Ring-𝔽 A B C g f) ＝
-    comp-hom-Ring-𝔽 A B D (comp-hom-Ring-𝔽 B C D h g) f
-  inv-associative-comp-hom-Ring-𝔽 =
-    inv-associative-comp-hom-Ring
-      ( ring-Ring-𝔽 A)
-      ( ring-Ring-𝔽 B)
-      ( ring-Ring-𝔽 C)
-      ( ring-Ring-𝔽 D)
-      ( h)
-      ( g)
-      ( f)
 ```
 
 ### Unit laws for composition of homomorphisms of commutative rings
diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
index cd43fb75bf..e34a655ed0 100644
--- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
+++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md
@@ -85,7 +85,7 @@ open import univalent-combinatorics.standard-finite-types
 ## Ideas
 
 The delooping of a group homomorphism `f : G → H` is a pointed map
-`Bf : BG → BH` equiped with a homotopy witnessing that the following square
+`Bf : BG → BH` equipped with a homotopy witnessing that the following square
 commutes :
 
 ```text
diff --git a/src/finite-group-theory/finite-abelian-groups.lagda.md b/src/finite-group-theory/finite-abelian-groups.lagda.md
index 137c9c647a..2b8dd899f6 100644
--- a/src/finite-group-theory/finite-abelian-groups.lagda.md
+++ b/src/finite-group-theory/finite-abelian-groups.lagda.md
@@ -115,7 +115,7 @@ module _
   semigroup-Ab-𝔽 : Semigroup l
   semigroup-Ab-𝔽 = semigroup-Group group-Ab-𝔽
 
-  is-group-Ab-𝔽 : is-group semigroup-Ab-𝔽
+  is-group-Ab-𝔽 : is-group-Semigroup semigroup-Ab-𝔽
   is-group-Ab-𝔽 = is-group-Group group-Ab-𝔽
 
   has-zero-Ab-𝔽 : is-unital-Semigroup semigroup-Ab-𝔽
@@ -139,7 +139,7 @@ module _
   right-unit-law-add-Ab-𝔽 : (x : type-Ab-𝔽) → add-Ab-𝔽 x zero-Ab-𝔽 ＝ x
   right-unit-law-add-Ab-𝔽 = right-unit-law-mul-Group group-Ab-𝔽
 
-  has-negatives-Ab-𝔽 : is-group' semigroup-Ab-𝔽 has-zero-Ab-𝔽
+  has-negatives-Ab-𝔽 : is-group-is-unital-Semigroup semigroup-Ab-𝔽 has-zero-Ab-𝔽
   has-negatives-Ab-𝔽 = has-inverses-Group group-Ab-𝔽
 
   neg-Ab-𝔽 : type-Ab-𝔽 → type-Ab-𝔽
diff --git a/src/finite-group-theory/finite-groups.lagda.md b/src/finite-group-theory/finite-groups.lagda.md
index fbb7a1421a..b19ed72587 100644
--- a/src/finite-group-theory/finite-groups.lagda.md
+++ b/src/finite-group-theory/finite-groups.lagda.md
@@ -16,7 +16,6 @@ open import foundation.binary-embeddings
 open import foundation.binary-equivalences
 open import foundation.decidable-equality
 open import foundation.decidable-types
-open import foundation.dependent-pair-types
 open import foundation.embeddings
 open import foundation.equivalences
 open import foundation.function-types
@@ -79,7 +78,7 @@ usual algebraic sense, i.e., it consists of a
 ```agda
 is-group-𝔽 :
   {l : Level} (G : Semigroup-𝔽 l) → UU l
-is-group-𝔽 G = is-group (semigroup-Semigroup-𝔽 G)
+is-group-𝔽 G = is-group-Semigroup (semigroup-Semigroup-𝔽 G)
 ```
 
 ### The type of finite groups
@@ -100,7 +99,7 @@ module _
   semigroup-Group-𝔽 =
     semigroup-Semigroup-𝔽 finite-semigroup-Group-𝔽
 
-  is-group-Group-𝔽 : is-group semigroup-Group-𝔽
+  is-group-Group-𝔽 : is-group-Semigroup semigroup-Group-𝔽
   is-group-Group-𝔽 = pr2 G
 
   group-Group-𝔽 : Group l
@@ -199,7 +198,7 @@ module _
   pointed-type-Group-𝔽 = pointed-type-Group group-Group-𝔽
 
   has-inverses-Group-𝔽 :
-    is-group' semigroup-Group-𝔽 is-unital-Group-𝔽
+    is-group-is-unital-Semigroup semigroup-Group-𝔽 is-unital-Group-𝔽
   has-inverses-Group-𝔽 = has-inverses-Group group-Group-𝔽
 
   inv-Group-𝔽 : type-Group-𝔽 → type-Group-𝔽
@@ -333,13 +332,13 @@ Group-of-Order l n = Σ (Group l) (λ G → mere-equiv (Fin n) (type-Group G))
 
 ## Properties
 
-### The type `is-group G` is finite for any semigroup of fixed finite order
+### The type `is-group-Semigroup G` is finite for any semigroup of fixed finite order
 
 ```agda
-is-finite-is-group :
+is-finite-is-group-Semigroup :
   {l : Level} (n : ℕ) (G : Semigroup-of-Order l n) →
-  is-finite {l} (is-group (pr1 G))
-is-finite-is-group {l} n G =
+  is-finite {l} (is-group-Semigroup (pr1 G))
+is-finite-is-group-Semigroup {l} n G =
   apply-universal-property-trunc-Prop
     ( pr2 G)
     ( is-finite-Prop _)
@@ -393,11 +392,11 @@ is-π-finite-Group-of-Order {l} k n =
       ( is-π-finite-Semigroup-of-Order (succ-ℕ k) n)
       ( λ X →
         is-π-finite-is-finite k
-          ( is-finite-is-group n X)))
+          ( is-finite-is-group-Semigroup n X)))
   where
   e :
     Group-of-Order l n ≃
-    Σ (Semigroup-of-Order l n) (λ X → is-group (pr1 X))
+    Σ (Semigroup-of-Order l n) (λ X → is-group-Semigroup (pr1 X))
   e = equiv-right-swap-Σ
 
 number-of-groups-of-order : ℕ → ℕ
diff --git a/src/foundation/category-of-sets.lagda.md b/src/foundation/category-of-sets.lagda.md
index 803e007a6a..b465639494 100644
--- a/src/foundation/category-of-sets.lagda.md
+++ b/src/foundation/category-of-sets.lagda.md
@@ -17,6 +17,7 @@ open import foundation.dependent-pair-types
 open import foundation.fundamental-theorem-of-identity-types
 open import foundation.isomorphisms-of-sets
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 
 open import foundation-core.contractible-types
@@ -44,8 +45,9 @@ obj-Large-Precategory Set-Large-Precategory = Set
 hom-set-Large-Precategory Set-Large-Precategory = hom-set-Set
 comp-hom-Large-Precategory Set-Large-Precategory g f = g ∘ f
 id-hom-Large-Precategory Set-Large-Precategory = id
-associative-comp-hom-Large-Precategory Set-Large-Precategory h g f = refl
-inv-associative-comp-hom-Large-Precategory Set-Large-Precategory h g f = refl
+involutive-eq-associative-comp-hom-Large-Precategory Set-Large-Precategory
+  h g f =
+  reflⁱ
 left-unit-law-comp-hom-Large-Precategory Set-Large-Precategory f = refl
 right-unit-law-comp-hom-Large-Precategory Set-Large-Precategory f = refl
 ```
diff --git a/src/foundation/large-locale-of-propositions.lagda.md b/src/foundation/large-locale-of-propositions.lagda.md
index faeb653117..233f214be3 100644
--- a/src/foundation/large-locale-of-propositions.lagda.md
+++ b/src/foundation/large-locale-of-propositions.lagda.md
@@ -38,7 +38,7 @@ form a [large locale](order-theory.large-locales.md).
 ### The large preorder of propositions
 
 ```agda
-Prop-Large-Preorder : Large-Preorder lsuc _⊔_
+Prop-Large-Preorder : Large-Preorder lsuc (_⊔_)
 type-Large-Preorder Prop-Large-Preorder = Prop
 leq-prop-Large-Preorder Prop-Large-Preorder = hom-Prop
 refl-leq-Large-Preorder Prop-Large-Preorder P = id
@@ -48,7 +48,7 @@ transitive-leq-Large-Preorder Prop-Large-Preorder P Q R g f = g ∘ f
 ### The large poset of propositions
 
 ```agda
-Prop-Large-Poset : Large-Poset lsuc _⊔_
+Prop-Large-Poset : Large-Poset lsuc (_⊔_)
 large-preorder-Large-Poset Prop-Large-Poset = Prop-Large-Preorder
 antisymmetric-leq-Large-Poset Prop-Large-Poset P Q = eq-iff
 ```
@@ -105,7 +105,7 @@ is-least-upper-bound-sup-has-least-upper-bound-family-of-elements-Large-Poset
 ### The large frame of propositions
 
 ```agda
-Prop-Large-Frame : Large-Frame lsuc _⊔_ lzero
+Prop-Large-Frame : Large-Frame lsuc (_⊔_) lzero
 large-poset-Large-Frame Prop-Large-Frame =
   Prop-Large-Poset
 is-large-meet-semilattice-Large-Frame Prop-Large-Frame =
@@ -119,6 +119,6 @@ distributive-meet-sup-Large-Frame Prop-Large-Frame =
 ### The large locale of propositions
 
 ```agda
-Prop-Large-Locale : Large-Locale lsuc _⊔_ lzero
+Prop-Large-Locale : Large-Locale lsuc (_⊔_) lzero
 Prop-Large-Locale = Prop-Large-Frame
 ```
diff --git a/src/foundation/mere-embeddings.lagda.md b/src/foundation/mere-embeddings.lagda.md
index f4947fe895..49197506fd 100644
--- a/src/foundation/mere-embeddings.lagda.md
+++ b/src/foundation/mere-embeddings.lagda.md
@@ -54,7 +54,7 @@ transitive-mere-emb g f =
         ( mere-emb-Prop _ _)
         ( λ f' → unit-trunc-Prop (comp-emb g' f')))
 
-mere-emb-Large-Preorder : Large-Preorder lsuc _⊔_
+mere-emb-Large-Preorder : Large-Preorder lsuc (_⊔_)
 type-Large-Preorder mere-emb-Large-Preorder l = UU l
 leq-prop-Large-Preorder mere-emb-Large-Preorder = mere-emb-Prop
 refl-leq-Large-Preorder mere-emb-Large-Preorder = refl-mere-emb
diff --git a/src/foundation/strictly-involutive-identity-types.lagda.md b/src/foundation/strictly-involutive-identity-types.lagda.md
index 0484a7fb0a..79e477f008 100644
--- a/src/foundation/strictly-involutive-identity-types.lagda.md
+++ b/src/foundation/strictly-involutive-identity-types.lagda.md
@@ -7,11 +7,14 @@ module foundation.strictly-involutive-identity-types where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.equality-cartesian-product-types
+open import foundation.function-extensionality
 open import foundation.multivariable-homotopies
 open import foundation.strictly-right-unital-concatenation-identifications
+open import foundation.univalence
 open import foundation.universal-property-identity-systems
 open import foundation.universe-levels
 
@@ -19,11 +22,14 @@ open import foundation-core.cartesian-product-types
 open import foundation-core.contractible-types
 open import foundation-core.equality-dependent-pair-types
 open import foundation-core.equivalences
+open import foundation-core.function-types
 open import foundation-core.functoriality-dependent-pair-types
+open import foundation-core.homotopies
 open import foundation-core.identity-types
 open import foundation-core.retractions
 open import foundation-core.sections
 open import foundation-core.torsorial-type-families
+open import foundation-core.transport-along-identifications
 ```
 
 </details>
@@ -538,6 +544,185 @@ module _
         ( left-unit-right-strict-concat))
 ```
 
+## Operations
+
+We define a range of basic operations on the strictly involutive identifications
+that all enjoy strict computational properties.
+
+### Action of functions on strictly involutive identifications
+
+```agda
+module _
+  {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B)
+  where
+
+  eq-ap-involutive-Id : {x y : A} → x ＝ⁱ y → f x ＝ f y
+  eq-ap-involutive-Id = ap f ∘ eq-involutive-eq
+
+  ap-involutive-Id : {x y : A} → x ＝ⁱ y → f x ＝ⁱ f y
+  ap-involutive-Id = involutive-eq-eq ∘ eq-ap-involutive-Id
+
+  compute-ap-refl-involutive-Id :
+    {x : A} → ap-involutive-Id (reflⁱ {x = x}) ＝ reflⁱ
+  compute-ap-refl-involutive-Id = refl
+
+module _
+  {l1 : Level} {A : UU l1}
+  where
+
+  compute-ap-id-involutive-Id :
+    {x y : A} (p : x ＝ⁱ y) → ap-involutive-Id id p ＝ p
+  compute-ap-id-involutive-Id (z , q , refl) =
+    eq-pair-eq-fiber (eq-pair (ap-id q) refl)
+```
+
+### Action of binary functions on strictly involutive identifications
+
+```agda
+module _
+  {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} (f : A → B → C)
+  where
+
+  ap-binary-involutive-Id :
+    {x x' : A} (p : x ＝ⁱ x') {y y' : B} (q : y ＝ⁱ y') → f x y ＝ⁱ f x' y'
+  ap-binary-involutive-Id (z , p1 , p2) (w , q1 , q2) =
+    ( f z w , ap-binary f p1 q1 , ap-binary f p2 q2)
+
+  left-unit-ap-binary-involutive-Id :
+    {x : A} {y y' : B} (q : y ＝ⁱ y') →
+    ap-binary-involutive-Id reflⁱ q ＝ ap-involutive-Id (f x) q
+  left-unit-ap-binary-involutive-Id (z , p , refl) = refl
+
+  right-unit-ap-binary-involutive-Id :
+    {x x' : A} (p : x ＝ⁱ x') {y : B} →
+    ap-binary-involutive-Id p reflⁱ ＝ ap-involutive-Id (λ z → f z y) p
+  right-unit-ap-binary-involutive-Id {.z} {x'} (z , p , refl) {y} =
+    eq-pair-eq-fiber (eq-pair right-unit refl)
+```
+
+### Transport along strictly involutive identifications
+
+```agda
+module _
+  {l1 l2 : Level} {A : UU l1} (B : A → UU l2)
+  where
+
+  tr-involutive-Id : {x y : A} → x ＝ⁱ y → B x → B y
+  tr-involutive-Id = tr B ∘ eq-involutive-eq
+
+  compute-tr-refl-involutive-Id :
+    {x : A} → tr-involutive-Id (reflⁱ {x = x}) ＝ id
+  compute-tr-refl-involutive-Id = refl
+```
+
+### Function extensionality with respect to strictly involutive identifications
+
+```agda
+module _
+  {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g : (x : A) → B x}
+  where
+
+  htpy-involutive-eq : f ＝ⁱ g → f ~ g
+  htpy-involutive-eq = htpy-eq ∘ eq-involutive-eq
+
+  involutive-eq-htpy : f ~ g → f ＝ⁱ g
+  involutive-eq-htpy = involutive-eq-eq ∘ eq-htpy
+
+  equiv-htpy-involutive-eq : (f ＝ⁱ g) ≃ (f ~ g)
+  equiv-htpy-involutive-eq = equiv-funext ∘e equiv-eq-involutive-eq
+
+  equiv-involutive-eq-htpy : (f ~ g) ≃ (f ＝ⁱ g)
+  equiv-involutive-eq-htpy = equiv-involutive-eq-eq ∘e equiv-eq-htpy
+
+  funext-involutive-Id : is-equiv htpy-involutive-eq
+  funext-involutive-Id = is-equiv-map-equiv equiv-htpy-involutive-eq
+```
+
+```agda
+module _
+  {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {f g : (x : A) → B x}
+  where
+
+  involutive-htpy-involutive-eq : f ＝ⁱ g → (x : A) → f x ＝ⁱ g x
+  involutive-htpy-involutive-eq (h , p , q) x =
+    ( h x , htpy-eq p x , htpy-eq q x)
+
+  involutive-eq-involutive-htpy : ((x : A) → f x ＝ⁱ g x) → f ＝ⁱ g
+  involutive-eq-involutive-htpy H =
+    ( pr1 ∘ H , eq-htpy (pr1 ∘ (pr2 ∘ H)) , eq-htpy (pr2 ∘ (pr2 ∘ H)))
+```
+
+It remains to show that these maps are part of an equivalence.
+
+### Standard univalence with respect to strictly involutive identifications
+
+```agda
+module _
+  {l1 : Level} {A B : UU l1}
+  where
+
+  map-involutive-eq : A ＝ⁱ B → A → B
+  map-involutive-eq = map-eq ∘ eq-involutive-eq
+
+  equiv-involutive-eq : A ＝ⁱ B → A ≃ B
+  equiv-involutive-eq = equiv-eq ∘ eq-involutive-eq
+
+  involutive-eq-equiv : A ≃ B → A ＝ⁱ B
+  involutive-eq-equiv = involutive-eq-eq ∘ eq-equiv
+
+  equiv-equiv-involutive-eq : (A ＝ⁱ B) ≃ (A ≃ B)
+  equiv-equiv-involutive-eq = equiv-univalence ∘e equiv-eq-involutive-eq
+
+  is-equiv-equiv-involutive-eq : is-equiv equiv-involutive-eq
+  is-equiv-equiv-involutive-eq = is-equiv-map-equiv equiv-equiv-involutive-eq
+
+  equiv-involutive-eq-equiv : (A ≃ B) ≃ (A ＝ⁱ B)
+  equiv-involutive-eq-equiv = equiv-involutive-eq-eq ∘e equiv-eq-equiv A B
+
+  is-equiv-involutive-eq-equiv : is-equiv involutive-eq-equiv
+  is-equiv-involutive-eq-equiv = is-equiv-map-equiv equiv-involutive-eq-equiv
+```
+
+### Whiskering of strictly involutive identifications
+
+```agda
+module _
+  {l : Level} {A : UU l} {x y z : A}
+  where
+
+  left-whisker-concat-involutive-Id :
+    (p : x ＝ⁱ y) {q r : y ＝ⁱ z} → q ＝ⁱ r → p ∙ⁱ q ＝ⁱ p ∙ⁱ r
+  left-whisker-concat-involutive-Id p β = ap-involutive-Id (p ∙ⁱ_) β
+
+  right-whisker-concat-involutive-Id :
+    {p q : x ＝ⁱ y} → p ＝ⁱ q → (r : y ＝ⁱ z) → p ∙ⁱ r ＝ⁱ q ∙ⁱ r
+  right-whisker-concat-involutive-Id α r = ap-involutive-Id (_∙ⁱ r) α
+```
+
+### Horizontal concatenation of strictly involutive identifications
+
+```agda
+module _
+  {l : Level} {A : UU l} {x y z : A}
+  where
+
+  horizontal-concat-involutive-Id² :
+    {p q : x ＝ⁱ y} → p ＝ⁱ q → {u v : y ＝ⁱ z} → u ＝ⁱ v → p ∙ⁱ u ＝ⁱ q ∙ⁱ v
+  horizontal-concat-involutive-Id² = ap-binary-involutive-Id (_∙ⁱ_)
+```
+
+### Vertical concatenation of strictly involutive identifications
+
+```agda
+module _
+  {l : Level} {A : UU l} {x y : A}
+  where
+
+  vertical-concat-involutive-Id² :
+    {p q r : x ＝ⁱ y} → p ＝ⁱ q → q ＝ⁱ r → p ＝ⁱ r
+  vertical-concat-involutive-Id² α β = α ∙ⁱ β
+```
+
 ## See also
 
 - [The Yoneda identity types](foundation.yoneda-identity-types.md) for an
diff --git a/src/foundation/yoneda-identity-types.lagda.md b/src/foundation/yoneda-identity-types.lagda.md
index 081408e0ba..c763b6c36a 100644
--- a/src/foundation/yoneda-identity-types.lagda.md
+++ b/src/foundation/yoneda-identity-types.lagda.md
@@ -7,7 +7,6 @@ module foundation.yoneda-identity-types where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.function-extensionality
@@ -719,7 +718,7 @@ on both arguments using one of the two sides in the Gray interchanger diagram
   ap (r ↦ f r y) p |                    | ap (r ↦ f r y') p
                    |                    |
                    ∨                    ∨
-                 f x' y ------------> f x' y'.
+                 f x' y ------------> f x' y'
                       ap (r ↦ f x' r) q
 ```
 
@@ -736,15 +735,15 @@ module _
   ap-binary-yoneda-Id {x} {x'} p {y} {y'} q =
     ap-yoneda-Id (λ z → f z y) p ∙ʸ ap-yoneda-Id (f x') q
 
-  left-unit-ap-binary-Id :
+  left-unit-ap-binary-yoneda-Id :
     {x : A} {y y' : B} (q : y ＝ʸ y') →
     ap-binary-yoneda-Id reflʸ q ＝ ap-yoneda-Id (f x) q
-  left-unit-ap-binary-Id q = refl
+  left-unit-ap-binary-yoneda-Id q = refl
 
-  right-unit-ap-binary-Id :
+  right-unit-ap-binary-yoneda-Id :
     {x x' : A} (p : x ＝ʸ x') {y : B} →
     ap-binary-yoneda-Id p reflʸ ＝ ap-yoneda-Id (λ z → f z y) p
-  right-unit-ap-binary-Id p = refl
+  right-unit-ap-binary-yoneda-Id p = refl
 ```
 
 ### Transport along Yoneda identifications
@@ -820,13 +819,13 @@ module _
   {l : Level} {A : UU l} {x y z : A}
   where
 
-  left-whisker-concat-yoenda-Id :
+  left-whisker-concat-yoneda-Id :
     (p : x ＝ʸ y) {q r : y ＝ʸ z} → q ＝ʸ r → p ∙ʸ q ＝ʸ p ∙ʸ r
-  left-whisker-concat-yoenda-Id p β = ap-yoneda-Id (p ∙ʸ_) β
+  left-whisker-concat-yoneda-Id p β = ap-yoneda-Id (p ∙ʸ_) β
 
-  right-whisker-concat-yoenda-Id :
+  right-whisker-concat-yoneda-Id :
     {p q : x ＝ʸ y} → p ＝ʸ q → (r : y ＝ʸ z) → p ∙ʸ r ＝ʸ q ∙ʸ r
-  right-whisker-concat-yoenda-Id α r = ap-yoneda-Id (_∙ʸ r) α
+  right-whisker-concat-yoneda-Id α r = ap-yoneda-Id (_∙ʸ r) α
 ```
 
 ### Horizontal concatenation of Yoneda identifications
@@ -847,13 +846,13 @@ module _
   compute-left-horizontal-concat-yoneda-Id² :
     {p : x ＝ʸ y} {u v : y ＝ʸ z} (β : u ＝ʸ v) →
     horizontal-concat-yoneda-Id² reflʸ β ＝
-    left-whisker-concat-yoenda-Id p β
+    left-whisker-concat-yoneda-Id p β
   compute-left-horizontal-concat-yoneda-Id² β = refl
 
   compute-right-horizontal-concat-yoneda-Id² :
     {p q : x ＝ʸ y} (α : p ＝ʸ q) {u : y ＝ʸ z} →
     horizontal-concat-yoneda-Id² α (reflʸ {x = u}) ＝
-    right-whisker-concat-yoenda-Id α u
+    right-whisker-concat-yoneda-Id α u
   compute-right-horizontal-concat-yoneda-Id² α = refl
 
 module _
diff --git a/src/group-theory.lagda.md b/src/group-theory.lagda.md
index e1be2d9b8d..5ce7c983b7 100644
--- a/src/group-theory.lagda.md
+++ b/src/group-theory.lagda.md
@@ -135,7 +135,6 @@ open import group-theory.opposite-semigroups public
 open import group-theory.orbit-stabilizer-theorem-concrete-groups public
 open import group-theory.orbits-concrete-group-actions public
 open import group-theory.orbits-group-actions public
-open import group-theory.orbits-monoid-actions public
 open import group-theory.orders-of-elements-groups public
 open import group-theory.perfect-cores public
 open import group-theory.perfect-groups public
@@ -148,6 +147,7 @@ open import group-theory.precategory-of-concrete-groups public
 open import group-theory.precategory-of-group-actions public
 open import group-theory.precategory-of-groups public
 open import group-theory.precategory-of-monoids public
+open import group-theory.precategory-of-orbits-monoid-actions public
 open import group-theory.precategory-of-semigroups public
 open import group-theory.principal-group-actions public
 open import group-theory.principal-torsors-concrete-groups public
diff --git a/src/group-theory/abelian-groups.lagda.md b/src/group-theory/abelian-groups.lagda.md
index 518213fe02..ab6e812b23 100644
--- a/src/group-theory/abelian-groups.lagda.md
+++ b/src/group-theory/abelian-groups.lagda.md
@@ -127,7 +127,7 @@ module _
   semigroup-Ab : Semigroup l
   semigroup-Ab = semigroup-Group group-Ab
 
-  is-group-Ab : is-group semigroup-Ab
+  is-group-Ab : is-group-Semigroup semigroup-Ab
   is-group-Ab = is-group-Group group-Ab
 
   has-zero-Ab : is-unital-Semigroup semigroup-Ab
@@ -154,7 +154,7 @@ module _
   right-unit-law-add-Ab : (x : type-Ab) → add-Ab x zero-Ab ＝ x
   right-unit-law-add-Ab = right-unit-law-mul-Group group-Ab
 
-  has-negatives-Ab : is-group' semigroup-Ab has-zero-Ab
+  has-negatives-Ab : is-group-is-unital-Semigroup semigroup-Ab has-zero-Ab
   has-negatives-Ab = has-inverses-Group group-Ab
 
   neg-Ab : type-Ab → type-Ab
diff --git a/src/group-theory/category-of-abelian-groups.lagda.md b/src/group-theory/category-of-abelian-groups.lagda.md
index 6f778a9f5a..abb1ac4341 100644
--- a/src/group-theory/category-of-abelian-groups.lagda.md
+++ b/src/group-theory/category-of-abelian-groups.lagda.md
@@ -10,13 +10,10 @@ module group-theory.category-of-abelian-groups where
 open import category-theory.categories
 open import category-theory.full-large-subcategories
 open import category-theory.functors-large-categories
-open import category-theory.functors-large-precategories
 open import category-theory.large-categories
 open import category-theory.large-precategories
 open import category-theory.precategories
 
-open import foundation.function-types
-open import foundation.identity-types
 open import foundation.universe-levels
 
 open import group-theory.abelian-groups
@@ -38,7 +35,7 @@ The **category of abelian groups** is the
 ### The large category of abelian groups
 
 ```agda
-Ab-Large-Category : Large-Category lsuc _⊔_
+Ab-Large-Category : Large-Category lsuc (_⊔_)
 Ab-Large-Category =
   large-category-Full-Large-Subcategory
     ( Group-Large-Category)
@@ -48,7 +45,7 @@ Ab-Large-Category =
 ### The large precategory of abelian groups
 
 ```agda
-Ab-Large-Precategory : Large-Precategory lsuc _⊔_
+Ab-Large-Precategory : Large-Precategory lsuc (_⊔_)
 Ab-Large-Precategory =
   large-precategory-Large-Category Ab-Large-Category
 ```
diff --git a/src/group-theory/category-of-groups.lagda.md b/src/group-theory/category-of-groups.lagda.md
index 6114c0e1c6..e543101e66 100644
--- a/src/group-theory/category-of-groups.lagda.md
+++ b/src/group-theory/category-of-groups.lagda.md
@@ -28,9 +28,7 @@ open import group-theory.precategory-of-groups
 is-large-category-Group :
   is-large-category-Large-Precategory Group-Large-Precategory
 is-large-category-Group G =
-  fundamental-theorem-id
-    ( is-torsorial-iso-Group G)
-    ( iso-eq-Group G)
+  fundamental-theorem-id (is-torsorial-iso-Group G) (iso-eq-Group G)
 
 eq-iso-Group : {l : Level} (G H : Group l) → iso-Group G H → Id G H
 eq-iso-Group G H = map-inv-is-equiv (is-large-category-Group G H)
diff --git a/src/group-theory/category-of-semigroups.lagda.md b/src/group-theory/category-of-semigroups.lagda.md
index ceb07b9051..997979a69e 100644
--- a/src/group-theory/category-of-semigroups.lagda.md
+++ b/src/group-theory/category-of-semigroups.lagda.md
@@ -36,9 +36,7 @@ category.
 is-large-category-Semigroup :
   is-large-category-Large-Precategory Semigroup-Large-Precategory
 is-large-category-Semigroup G =
-  fundamental-theorem-id
-    ( is-torsorial-iso-Semigroup G)
-    ( iso-eq-Semigroup G)
+  fundamental-theorem-id (is-torsorial-iso-Semigroup G) (iso-eq-Semigroup G)
 
 extensionality-Semigroup :
   {l : Level} (G H : Semigroup l) → Id G H ≃ iso-Semigroup G H
diff --git a/src/group-theory/commutative-monoids.lagda.md b/src/group-theory/commutative-monoids.lagda.md
index 61fd30c897..e48be716ac 100644
--- a/src/group-theory/commutative-monoids.lagda.md
+++ b/src/group-theory/commutative-monoids.lagda.md
@@ -11,6 +11,7 @@ open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.interchange-law
+open import foundation.iterated-dependent-product-types
 open import foundation.propositions
 open import foundation.sets
 open import foundation.unital-binary-operations
@@ -24,18 +25,36 @@ open import group-theory.semigroups
 
 ## Idea
 
-A commutative monoid is a monoid `M` in which `xy = yx` holds for all `x y : M`.
+A {{#concept "commutative monoid" Agda=Commutative-Monoid}} is a
+[monoid](group-theory.monoids.md) `M` in which `xy = yx` holds for all
+`x y : M`.
 
-## Definition
+## Definitions
 
-### Commutative monoids
+### The predicate on monoids of being commutative
 
 ```agda
-is-commutative-Monoid :
-  {l : Level} (M : Monoid l) → UU l
-is-commutative-Monoid M =
-  (x y : type-Monoid M) → Id (mul-Monoid M x y) (mul-Monoid M y x)
+module _
+  {l : Level} (M : Monoid l)
+  where
+
+  is-commutative-Monoid : UU l
+  is-commutative-Monoid =
+    (x y : type-Monoid M) → mul-Monoid M x y ＝ mul-Monoid M y x
+
+  is-prop-is-commutative-Monoid : is-prop is-commutative-Monoid
+  is-prop-is-commutative-Monoid =
+    is-prop-iterated-Π 2
+      ( λ x y → is-set-type-Monoid M (mul-Monoid M x y) (mul-Monoid M y x))
 
+  is-commutative-prop-Monoid : Prop l
+  is-commutative-prop-Monoid =
+    ( is-commutative-Monoid , is-prop-is-commutative-Monoid)
+```
+
+### Commutative monoids
+
+```agda
 Commutative-Monoid : (l : Level) → UU (lsuc l)
 Commutative-Monoid l = Σ (Monoid l) is-commutative-Monoid
 
diff --git a/src/group-theory/concrete-groups.lagda.md b/src/group-theory/concrete-groups.lagda.md
index 01941b4f68..0dfca3da67 100644
--- a/src/group-theory/concrete-groups.lagda.md
+++ b/src/group-theory/concrete-groups.lagda.md
@@ -184,14 +184,16 @@ module _
     right-unit-law-mul-Concrete-Group
 
   is-group-Concrete-Group' :
-    is-group' semigroup-Concrete-Group is-unital-semigroup-Concrete-Group
+    is-group-is-unital-Semigroup
+      ( semigroup-Concrete-Group)
+      ( is-unital-semigroup-Concrete-Group)
   pr1 is-group-Concrete-Group' = inv-Concrete-Group
   pr1 (pr2 is-group-Concrete-Group') =
     left-inverse-law-mul-Concrete-Group
   pr2 (pr2 is-group-Concrete-Group') =
     right-inverse-law-mul-Concrete-Group
 
-  is-group-Concrete-Group : is-group semigroup-Concrete-Group
+  is-group-Concrete-Group : is-group-Semigroup semigroup-Concrete-Group
   pr1 is-group-Concrete-Group = is-unital-semigroup-Concrete-Group
   pr2 is-group-Concrete-Group = is-group-Concrete-Group'
 
@@ -215,7 +217,7 @@ module _
   is-unital-op-semigroup-Concrete-Group =
     is-unital-op-Group (group-Concrete-Group G)
 
-  is-group-op-Concrete-Group : is-group op-semigroup-Concrete-Group
+  is-group-op-Concrete-Group : is-group-Semigroup op-semigroup-Concrete-Group
   is-group-op-Concrete-Group = is-group-op-Group (group-Concrete-Group G)
 
   op-group-Concrete-Group : Group l
diff --git a/src/group-theory/cores-monoids.lagda.md b/src/group-theory/cores-monoids.lagda.md
index ab9d3a1329..50328b33a1 100644
--- a/src/group-theory/cores-monoids.lagda.md
+++ b/src/group-theory/cores-monoids.lagda.md
@@ -111,12 +111,13 @@ module _
       ( subtype-core-Monoid)
       ( is-right-inverse-inv-is-invertible-element-Monoid M (pr2 x))
 
-  is-group-core-Monoid' : is-group' semigroup-core-Monoid is-unital-core-Monoid
+  is-group-core-Monoid' :
+    is-group-is-unital-Semigroup semigroup-core-Monoid is-unital-core-Monoid
   pr1 is-group-core-Monoid' = inv-core-Monoid
   pr1 (pr2 is-group-core-Monoid') = left-inverse-law-mul-core-Monoid
   pr2 (pr2 is-group-core-Monoid') = right-inverse-law-mul-core-Monoid
 
-  is-group-core-Monoid : is-group semigroup-core-Monoid
+  is-group-core-Monoid : is-group-Semigroup semigroup-core-Monoid
   pr1 is-group-core-Monoid = is-unital-core-Monoid
   pr2 is-group-core-Monoid = is-group-core-Monoid'
 
diff --git a/src/group-theory/dependent-products-abelian-groups.lagda.md b/src/group-theory/dependent-products-abelian-groups.lagda.md
index 77af46cc8d..94aa816e80 100644
--- a/src/group-theory/dependent-products-abelian-groups.lagda.md
+++ b/src/group-theory/dependent-products-abelian-groups.lagda.md
@@ -82,7 +82,7 @@ module _
     (f : type-Π-Ab) → add-Π-Ab f (neg-Π-Ab f) ＝ zero-Π-Ab
   right-inverse-law-add-Π-Ab = right-inverse-law-mul-Group group-Π-Ab
 
-  is-group-Π-Ab : is-group semigroup-Π-Ab
+  is-group-Π-Ab : is-group-Semigroup semigroup-Π-Ab
   is-group-Π-Ab = is-group-Group group-Π-Ab
 
   commutative-add-Π-Ab :
diff --git a/src/group-theory/dependent-products-groups.lagda.md b/src/group-theory/dependent-products-groups.lagda.md
index 80997e68b6..6c0d59730a 100644
--- a/src/group-theory/dependent-products-groups.lagda.md
+++ b/src/group-theory/dependent-products-groups.lagda.md
@@ -91,7 +91,7 @@ module _
   right-inverse-law-mul-Π-Group f =
     eq-htpy (λ x → right-inverse-law-mul-Group (G x) (f x))
 
-  is-group-Π-Group : is-group semigroup-Π-Group
+  is-group-Π-Group : is-group-Semigroup semigroup-Π-Group
   pr1 is-group-Π-Group = is-unital-Π-Group
   pr1 (pr2 is-group-Π-Group) = inv-Π-Group
   pr1 (pr2 (pr2 is-group-Π-Group)) = left-inverse-law-mul-Π-Group
diff --git a/src/group-theory/function-groups.lagda.md b/src/group-theory/function-groups.lagda.md
index c3dcabbead..b9d8cff7aa 100644
--- a/src/group-theory/function-groups.lagda.md
+++ b/src/group-theory/function-groups.lagda.md
@@ -94,6 +94,6 @@ module _
   right-inverse-law-mul-function-Group =
     right-inverse-law-mul-Π-Group X (λ _ → G)
 
-  is-group-function-Group : is-group semigroup-function-Group
+  is-group-function-Group : is-group-Semigroup semigroup-function-Group
   is-group-function-Group = is-group-Π-Group X (λ _ → G)
 ```
diff --git a/src/group-theory/groups.lagda.md b/src/group-theory/groups.lagda.md
index 513db7f496..3aa3bee992 100644
--- a/src/group-theory/groups.lagda.md
+++ b/src/group-theory/groups.lagda.md
@@ -59,9 +59,9 @@ an inverse operation `x ↦ x⁻¹` satisfying the group laws
 ### The condition that a semigroup is a group
 
 ```agda
-is-group' :
+is-group-is-unital-Semigroup :
   {l : Level} (G : Semigroup l) → is-unital-Semigroup G → UU l
-is-group' G is-unital-Semigroup-G =
+is-group-is-unital-Semigroup G is-unital-Semigroup-G =
   Σ ( type-Semigroup G → type-Semigroup G)
     ( λ i →
       ( (x : type-Semigroup G) →
@@ -69,10 +69,10 @@ is-group' G is-unital-Semigroup-G =
       ( (x : type-Semigroup G) →
         Id (mul-Semigroup G x (i x)) (pr1 is-unital-Semigroup-G)))
 
-is-group :
+is-group-Semigroup :
   {l : Level} (G : Semigroup l) → UU l
-is-group G =
-  Σ (is-unital-Semigroup G) (is-group' G)
+is-group-Semigroup G =
+  Σ (is-unital-Semigroup G) (is-group-is-unital-Semigroup G)
 ```
 
 ### The type of groups
@@ -80,7 +80,7 @@ is-group G =
 ```agda
 Group :
   (l : Level) → UU (lsuc l)
-Group l = Σ (Semigroup l) is-group
+Group l = Σ (Semigroup l) is-group-Semigroup
 
 module _
   {l : Level} (G : Group l)
@@ -117,7 +117,7 @@ module _
     Id (mul-Group (mul-Group x y) z) (mul-Group x (mul-Group y z))
   associative-mul-Group = pr2 has-associative-mul-Group
 
-  is-group-Group : is-group semigroup-Group
+  is-group-Group : is-group-Semigroup semigroup-Group
   is-group-Group = pr2 G
 
   is-unital-Group : is-unital-Semigroup semigroup-Group
@@ -162,7 +162,8 @@ module _
   pr1 pointed-type-Group = type-Group
   pr2 pointed-type-Group = unit-Group
 
-  has-inverses-Group : is-group' semigroup-Group is-unital-Group
+  has-inverses-Group :
+    is-group-is-unital-Semigroup semigroup-Group is-unital-Group
   has-inverses-Group = pr2 is-group-Group
 
   inv-Group : type-Group → type-Group
@@ -216,7 +217,8 @@ module _
 structure-group :
   {l1 : Level} → UU l1 → UU l1
 structure-group X =
-  Σ (structure-semigroup X) (λ p → is-group (semigroup-structure-semigroup X p))
+  Σ ( structure-semigroup X)
+    ( λ p → is-group-Semigroup (semigroup-structure-semigroup X p))
 
 group-structure-group :
   {l1 : Level} → (X : UU l1) → structure-group X → Group l1
@@ -545,10 +547,10 @@ module _
 
 ```agda
 abstract
-  all-elements-equal-is-group :
+  all-elements-equal-is-group-Semigroup :
     {l : Level} (G : Semigroup l) (e : is-unital-Semigroup G) →
-    all-elements-equal (is-group' G e)
-  all-elements-equal-is-group
+    all-elements-equal (is-group-is-unital-Semigroup G e)
+  all-elements-equal-is-group-Semigroup
     ( pair G (pair μ associative-G))
     ( pair e (pair left-unit-G right-unit-G))
     ( pair i (pair left-inv-i right-inv-i))
@@ -559,7 +561,8 @@ abstract
           ( Π-Prop (type-Set G) (λ x → Id-Prop G (μ (i x) x) e))
           ( Π-Prop (type-Set G) (λ x → Id-Prop G (μ x (i x)) e)))
       ( eq-htpy
-        ( λ x → equational-reasoning
+        ( λ x →
+          equational-reasoning
           i x
           ＝ μ e (i x)
             by inv (left-unit-G (i x))
@@ -573,17 +576,17 @@ abstract
             by right-unit-G (i' x)))
 
 abstract
-  is-prop-is-group :
-    {l : Level} (G : Semigroup l) → is-prop (is-group G)
-  is-prop-is-group G =
+  is-prop-is-group-Semigroup :
+    {l : Level} (G : Semigroup l) → is-prop (is-group-Semigroup G)
+  is-prop-is-group-Semigroup G =
     is-prop-Σ
       ( is-prop-is-unital-Semigroup G)
       ( λ e →
-        is-prop-all-elements-equal (all-elements-equal-is-group G e))
+        is-prop-all-elements-equal (all-elements-equal-is-group-Semigroup G e))
 
 is-group-prop-Semigroup : {l : Level} (G : Semigroup l) → Prop l
-pr1 (is-group-prop-Semigroup G) = is-group G
-pr2 (is-group-prop-Semigroup G) = is-prop-is-group G
+pr1 (is-group-prop-Semigroup G) = is-group-Semigroup G
+pr2 (is-group-prop-Semigroup G) = is-prop-is-group-Semigroup G
 ```
 
 ### Any idempotent element in a group is the unit
diff --git a/src/group-theory/homomorphisms-abelian-groups.lagda.md b/src/group-theory/homomorphisms-abelian-groups.lagda.md
index 9e5cac9313..172b824ed0 100644
--- a/src/group-theory/homomorphisms-abelian-groups.lagda.md
+++ b/src/group-theory/homomorphisms-abelian-groups.lagda.md
@@ -178,19 +178,6 @@ associative-comp-hom-Ab A B C D =
     ( semigroup-Ab B)
     ( semigroup-Ab C)
     ( semigroup-Ab D)
-
-inv-associative-comp-hom-Ab :
-  {l1 l2 l3 l4 : Level}
-  (A : Ab l1) (B : Ab l2) (C : Ab l3) (D : Ab l4)
-  (h : hom-Ab C D) (g : hom-Ab B C) (f : hom-Ab A B) →
-  comp-hom-Ab A C D h (comp-hom-Ab A B C g f) ＝
-  comp-hom-Ab A B D (comp-hom-Ab B C D h g) f
-inv-associative-comp-hom-Ab A B C D =
-  inv-associative-comp-hom-Semigroup
-    ( semigroup-Ab A)
-    ( semigroup-Ab B)
-    ( semigroup-Ab C)
-    ( semigroup-Ab D)
 ```
 
 ### The unit laws for composition of abelian groups
diff --git a/src/group-theory/homomorphisms-commutative-monoids.lagda.md b/src/group-theory/homomorphisms-commutative-monoids.lagda.md
index 998509491b..b5783ad749 100644
--- a/src/group-theory/homomorphisms-commutative-monoids.lagda.md
+++ b/src/group-theory/homomorphisms-commutative-monoids.lagda.md
@@ -222,23 +222,6 @@ module _
       ( monoid-Commutative-Monoid L)
       ( monoid-Commutative-Monoid M)
       ( monoid-Commutative-Monoid N)
-
-  inv-associative-comp-hom-Commutative-Monoid :
-    (h : hom-Commutative-Monoid M N)
-    (g : hom-Commutative-Monoid L M)
-    (f : hom-Commutative-Monoid K L) →
-    comp-hom-Commutative-Monoid K M N
-      ( h)
-      ( comp-hom-Commutative-Monoid K L M g f) ＝
-    comp-hom-Commutative-Monoid K L N
-      ( comp-hom-Commutative-Monoid L M N h g)
-      ( f)
-  inv-associative-comp-hom-Commutative-Monoid =
-    inv-associative-comp-hom-Monoid
-      ( monoid-Commutative-Monoid K)
-      ( monoid-Commutative-Monoid L)
-      ( monoid-Commutative-Monoid M)
-      ( monoid-Commutative-Monoid N)
 ```
 
 ### The unit laws for composition of homomorphisms of commutative monoids
diff --git a/src/group-theory/homomorphisms-concrete-groups.lagda.md b/src/group-theory/homomorphisms-concrete-groups.lagda.md
index 1ecb7ad778..d61de489f6 100644
--- a/src/group-theory/homomorphisms-concrete-groups.lagda.md
+++ b/src/group-theory/homomorphisms-concrete-groups.lagda.md
@@ -172,22 +172,6 @@ associative-comp-hom-Concrete-Group G H K L =
     ( ∞-group-Concrete-Group K)
     ( ∞-group-Concrete-Group L)
 
-inv-associative-comp-hom-Concrete-Group :
-  {l1 l2 l3 l4 : Level}
-  (G : Concrete-Group l1) (H : Concrete-Group l2)
-  (K : Concrete-Group l3) (L : Concrete-Group l4)
-  (h : hom-Concrete-Group K L) (g : hom-Concrete-Group H K)
-  (f : hom-Concrete-Group G H) →
-  htpy-hom-Concrete-Group G L
-    ( comp-hom-Concrete-Group G K L h (comp-hom-Concrete-Group G H K g f))
-    ( comp-hom-Concrete-Group G H L (comp-hom-Concrete-Group H K L h g) f)
-inv-associative-comp-hom-Concrete-Group G H K L =
-  inv-associative-comp-hom-∞-Group
-    ( ∞-group-Concrete-Group G)
-    ( ∞-group-Concrete-Group H)
-    ( ∞-group-Concrete-Group K)
-    ( ∞-group-Concrete-Group L)
-
 module _
   {l1 l2 : Level} (G : Concrete-Group l1) (H : Concrete-Group l2)
   where
diff --git a/src/group-theory/homomorphisms-group-actions.lagda.md b/src/group-theory/homomorphisms-group-actions.lagda.md
index e8f45751af..d615116603 100644
--- a/src/group-theory/homomorphisms-group-actions.lagda.md
+++ b/src/group-theory/homomorphisms-group-actions.lagda.md
@@ -256,19 +256,6 @@ module _
         ( h)
         ( comp-hom-action-Group G X1 X2 X3 g f))
       ( refl-htpy)
-
-  inv-associative-comp-hom-action-Group :
-    comp-hom-action-Group G X1 X3 X4 h (comp-hom-action-Group G X1 X2 X3 g f) ＝
-    comp-hom-action-Group G X1 X2 X4 (comp-hom-action-Group G X2 X3 X4 h g) f
-  inv-associative-comp-hom-action-Group =
-    eq-htpy-hom-action-Group G X1 X4
-      ( comp-hom-action-Group G X1 X3 X4
-        ( h)
-        ( comp-hom-action-Group G X1 X2 X3 g f))
-      ( comp-hom-action-Group G X1 X2 X4
-        ( comp-hom-action-Group G X2 X3 X4 h g)
-        ( f))
-      ( refl-htpy)
 ```
 
 ### Composition satisfies the left and right unit laws
diff --git a/src/group-theory/homomorphisms-groups.lagda.md b/src/group-theory/homomorphisms-groups.lagda.md
index 7f6ba599f2..a4e286a07f 100644
--- a/src/group-theory/homomorphisms-groups.lagda.md
+++ b/src/group-theory/homomorphisms-groups.lagda.md
@@ -192,17 +192,6 @@ module _
       ( semigroup-Group H)
       ( semigroup-Group K)
       ( semigroup-Group L)
-
-  inv-associative-comp-hom-Group :
-    (h : hom-Group K L) (g : hom-Group H K) (f : hom-Group G H) →
-    comp-hom-Group G K L h (comp-hom-Group G H K g f) ＝
-    comp-hom-Group G H L (comp-hom-Group H K L h g) f
-  inv-associative-comp-hom-Group =
-    inv-associative-comp-hom-Semigroup
-      ( semigroup-Group G)
-      ( semigroup-Group H)
-      ( semigroup-Group K)
-      ( semigroup-Group L)
 ```
 
 ### The left and right unit laws for composition of group homomorphisms
diff --git a/src/group-theory/homomorphisms-monoids.lagda.md b/src/group-theory/homomorphisms-monoids.lagda.md
index 5b647d2956..f8c37265bd 100644
--- a/src/group-theory/homomorphisms-monoids.lagda.md
+++ b/src/group-theory/homomorphisms-monoids.lagda.md
@@ -66,6 +66,13 @@ module _
   is-prop-preserves-unit-hom-Semigroup f =
     is-prop-type-Prop (preserves-unit-prop-hom-Semigroup f)
 
+  preserves-unit-hom-prop-Semigroup :
+    hom-Semigroup (semigroup-Monoid M1) (semigroup-Monoid M2) →
+    Prop l2
+  preserves-unit-hom-prop-Semigroup f =
+    ( preserves-unit-hom-Semigroup f ,
+      is-prop-preserves-unit-hom-Semigroup f)
+
   hom-set-Monoid : Set (l1 ⊔ l2)
   hom-set-Monoid =
     set-subset
@@ -234,16 +241,6 @@ module _
       ( comp-hom-Monoid K L N (comp-hom-Monoid L M N h g) f)
       ( comp-hom-Monoid K M N h (comp-hom-Monoid K L M g f))
       ( refl-htpy)
-
-  inv-associative-comp-hom-Monoid :
-    (h : hom-Monoid M N) (g : hom-Monoid L M) (f : hom-Monoid K L) →
-    comp-hom-Monoid K M N h (comp-hom-Monoid K L M g f) ＝
-    comp-hom-Monoid K L N (comp-hom-Monoid L M N h g) f
-  inv-associative-comp-hom-Monoid h g f =
-    eq-htpy-hom-Monoid K N
-      ( comp-hom-Monoid K M N h (comp-hom-Monoid K L M g f))
-      ( comp-hom-Monoid K L N (comp-hom-Monoid L M N h g) f)
-      ( refl-htpy)
 ```
 
 ### Unit laws for composition of homomorphisms of monoids
diff --git a/src/group-theory/homomorphisms-semigroups.lagda.md b/src/group-theory/homomorphisms-semigroups.lagda.md
index 53dc30b3c5..b00a30a3fa 100644
--- a/src/group-theory/homomorphisms-semigroups.lagda.md
+++ b/src/group-theory/homomorphisms-semigroups.lagda.md
@@ -216,11 +216,6 @@ module _
     comp-hom-Semigroup G H L (comp-hom-Semigroup H K L h g) f ＝
     comp-hom-Semigroup G K L h (comp-hom-Semigroup G H K g f)
   associative-comp-hom-Semigroup = eq-htpy-hom-Semigroup G L refl-htpy
-
-  inv-associative-comp-hom-Semigroup :
-    comp-hom-Semigroup G K L h (comp-hom-Semigroup G H K g f) ＝
-    comp-hom-Semigroup G H L (comp-hom-Semigroup H K L h g) f
-  inv-associative-comp-hom-Semigroup = eq-htpy-hom-Semigroup G L refl-htpy
 ```
 
 ### The left and right unit laws for composition of homomorphisms of semigroups
diff --git a/src/group-theory/loop-groups-sets.lagda.md b/src/group-theory/loop-groups-sets.lagda.md
index 098f85479c..15fdf92516 100644
--- a/src/group-theory/loop-groups-sets.lagda.md
+++ b/src/group-theory/loop-groups-sets.lagda.md
@@ -69,7 +69,9 @@ module _
   pr2 (pr2 is-unital-Semigroup-loop-semigroup-Set) x = right-unit
 
   is-group-loop-semigroup-Set' :
-    is-group' loop-semigroup-Set is-unital-Semigroup-loop-semigroup-Set
+    is-group-is-unital-Semigroup
+      ( loop-semigroup-Set)
+      ( is-unital-Semigroup-loop-semigroup-Set)
   pr1 is-group-loop-semigroup-Set' = inv
   pr1 (pr2 is-group-loop-semigroup-Set') = left-inv
   pr2 (pr2 is-group-loop-semigroup-Set') = right-inv
diff --git a/src/group-theory/opposite-groups.lagda.md b/src/group-theory/opposite-groups.lagda.md
index c8cb5e7bab..1b8b84aac2 100644
--- a/src/group-theory/opposite-groups.lagda.md
+++ b/src/group-theory/opposite-groups.lagda.md
@@ -36,7 +36,7 @@ module _
   pr1 (pr2 is-unital-op-Group) = right-unit-law-mul-Group G
   pr2 (pr2 is-unital-op-Group) = left-unit-law-mul-Group G
 
-  is-group-op-Group : is-group (op-Semigroup (semigroup-Group G))
+  is-group-op-Group : is-group-Semigroup (op-Semigroup (semigroup-Group G))
   pr1 is-group-op-Group = is-unital-op-Group
   pr1 (pr2 is-group-op-Group) = inv-Group G
   pr1 (pr2 (pr2 is-group-op-Group)) = right-inverse-law-mul-Group G
diff --git a/src/group-theory/precategory-of-commutative-monoids.lagda.md b/src/group-theory/precategory-of-commutative-monoids.lagda.md
index cb959988f3..f3d5757c93 100644
--- a/src/group-theory/precategory-of-commutative-monoids.lagda.md
+++ b/src/group-theory/precategory-of-commutative-monoids.lagda.md
@@ -7,12 +7,14 @@ module group-theory.precategory-of-commutative-monoids where
 <details><summary>Imports</summary>
 
 ```agda
+open import category-theory.full-large-subprecategories
 open import category-theory.large-precategories
+open import category-theory.precategories
 
 open import foundation.universe-levels
 
 open import group-theory.commutative-monoids
-open import group-theory.homomorphisms-commutative-monoids
+open import group-theory.precategory-of-monoids
 ```
 
 </details>
@@ -24,32 +26,28 @@ homomorphisms of monoids.
 
 ## Definitions
 
+### The precategory of commutative monoids as a full subprecategory of monoids
+
+```agda
+Commutative-Monoid-Full-Large-Subprecategory :
+  Full-Large-Subprecategory (λ l → l) Monoid-Large-Precategory
+Commutative-Monoid-Full-Large-Subprecategory = is-commutative-prop-Monoid
+```
+
 ### The large precategory of commutative monoids
 
 ```agda
-Commutative-Monoid-Large-Precategory : Large-Precategory lsuc _⊔_
-obj-Large-Precategory
-  Commutative-Monoid-Large-Precategory =
-  Commutative-Monoid
-hom-set-Large-Precategory
-  Commutative-Monoid-Large-Precategory =
-  hom-set-Commutative-Monoid
-comp-hom-Large-Precategory
-  Commutative-Monoid-Large-Precategory {X = K} {L} {M} =
-  comp-hom-Commutative-Monoid K L M
-id-hom-Large-Precategory
-  Commutative-Monoid-Large-Precategory {X = M} =
-  id-hom-Commutative-Monoid M
-associative-comp-hom-Large-Precategory
-  Commutative-Monoid-Large-Precategory {X = K} {L} {M} {N} =
-  associative-comp-hom-Commutative-Monoid K L M N
-inv-associative-comp-hom-Large-Precategory
-  Commutative-Monoid-Large-Precategory {X = K} {L} {M} {N} =
-  inv-associative-comp-hom-Commutative-Monoid K L M N
-left-unit-law-comp-hom-Large-Precategory
-  Commutative-Monoid-Large-Precategory {X = M} {N} =
-  left-unit-law-comp-hom-Commutative-Monoid M N
-right-unit-law-comp-hom-Large-Precategory
-  Commutative-Monoid-Large-Precategory {X = M} {N} =
-  right-unit-law-comp-hom-Commutative-Monoid M N
+Commutative-Monoid-Large-Precategory : Large-Precategory lsuc (_⊔_)
+Commutative-Monoid-Large-Precategory =
+  large-precategory-Full-Large-Subprecategory
+    ( Monoid-Large-Precategory)
+    ( Commutative-Monoid-Full-Large-Subprecategory)
+```
+
+### The precategory of small commutative monoids
+
+```agda
+Commutative-Monoid-Precategory : (l : Level) → Precategory (lsuc l) l
+Commutative-Monoid-Precategory =
+  precategory-Large-Precategory Commutative-Monoid-Large-Precategory
 ```
diff --git a/src/group-theory/precategory-of-concrete-groups.lagda.md b/src/group-theory/precategory-of-concrete-groups.lagda.md
index 2da61df681..43b955bd5d 100644
--- a/src/group-theory/precategory-of-concrete-groups.lagda.md
+++ b/src/group-theory/precategory-of-concrete-groups.lagda.md
@@ -23,32 +23,19 @@ open import group-theory.homomorphisms-concrete-groups
 
 ```agda
 Concrete-Group-Large-Precategory : Large-Precategory lsuc (_⊔_)
-obj-Large-Precategory
-  Concrete-Group-Large-Precategory =
-  Concrete-Group
-hom-set-Large-Precategory
-  Concrete-Group-Large-Precategory =
-  hom-set-Concrete-Group
-comp-hom-Large-Precategory
-  Concrete-Group-Large-Precategory {X = G} {Y = H} {Z = K} =
-  comp-hom-Concrete-Group G H K
-id-hom-Large-Precategory
-  Concrete-Group-Large-Precategory {X = G} =
-  id-hom-Concrete-Group G
-associative-comp-hom-Large-Precategory
-  Concrete-Group-Large-Precategory {X = G} {Y = H} {Z = K} {W = L} h g f =
-  eq-htpy-hom-Concrete-Group G L _ _
-    ( associative-comp-hom-Concrete-Group G H K L h g f)
-inv-associative-comp-hom-Large-Precategory
-  Concrete-Group-Large-Precategory {X = G} {Y = H} {Z = K} {W = L} h g f =
-  eq-htpy-hom-Concrete-Group G L _ _
-    ( inv-associative-comp-hom-Concrete-Group G H K L h g f)
-left-unit-law-comp-hom-Large-Precategory
-  Concrete-Group-Large-Precategory {X = G} {Y = H} f =
-  eq-htpy-hom-Concrete-Group G H _ _
-    ( left-unit-law-comp-hom-Concrete-Group G H f)
-right-unit-law-comp-hom-Large-Precategory
-  Concrete-Group-Large-Precategory {X = G} {Y = H} f =
-  eq-htpy-hom-Concrete-Group G H _ _
-    ( right-unit-law-comp-hom-Concrete-Group G H f)
+Concrete-Group-Large-Precategory =
+  make-Large-Precategory
+    ( Concrete-Group)
+    ( hom-set-Concrete-Group)
+    ( λ {l1} {l2} {l3} {G} {H} {K} → comp-hom-Concrete-Group G H K)
+    ( λ {l} {G} → id-hom-Concrete-Group G)
+    ( λ {l1} {l2} {l3} {l4} {G} {H} {K} {L} h g f →
+      eq-htpy-hom-Concrete-Group G L _ _
+        ( associative-comp-hom-Concrete-Group G H K L h g f))
+    ( λ {l1} {l2} {G} {H} f →
+      eq-htpy-hom-Concrete-Group G H _ _
+        ( left-unit-law-comp-hom-Concrete-Group G H f))
+    ( λ {l1} {l2} {G} {H} f →
+      eq-htpy-hom-Concrete-Group G H _ _
+        ( right-unit-law-comp-hom-Concrete-Group G H f))
 ```
diff --git a/src/group-theory/precategory-of-group-actions.lagda.md b/src/group-theory/precategory-of-group-actions.lagda.md
index 3ea66e7300..d8fc00a1c5 100644
--- a/src/group-theory/precategory-of-group-actions.lagda.md
+++ b/src/group-theory/precategory-of-group-actions.lagda.md
@@ -10,8 +10,6 @@ module group-theory.precategory-of-group-actions where
 open import category-theory.large-precategories
 open import category-theory.precategories
 
-open import foundation.dependent-pair-types
-open import foundation.identity-types
 open import foundation.universe-levels
 
 open import group-theory.group-actions
@@ -39,30 +37,16 @@ module _
 
   action-Group-Large-Precategory :
     Large-Precategory (λ l2 → l1 ⊔ lsuc l2) (λ l2 l3 → l1 ⊔ l2 ⊔ l3)
-  obj-Large-Precategory
-    action-Group-Large-Precategory =
-    action-Group G
-  hom-set-Large-Precategory
-    action-Group-Large-Precategory =
-    hom-set-action-Group G
-  comp-hom-Large-Precategory
-    action-Group-Large-Precategory {X = X} {Y} {Z} =
-    comp-hom-action-Group G X Y Z
-  id-hom-Large-Precategory
-    action-Group-Large-Precategory {X = X} =
-    id-hom-action-Group G X
-  associative-comp-hom-Large-Precategory
-    action-Group-Large-Precategory {X = X} {Y} {Z} {W} =
-    associative-comp-hom-action-Group G X Y Z W
-  inv-associative-comp-hom-Large-Precategory
-    action-Group-Large-Precategory {X = X} {Y} {Z} {W} =
-    inv-associative-comp-hom-action-Group G X Y Z W
-  left-unit-law-comp-hom-Large-Precategory
-    action-Group-Large-Precategory {X = X} {Y} =
-    left-unit-law-comp-hom-action-Group G X Y
-  right-unit-law-comp-hom-Large-Precategory
-    action-Group-Large-Precategory {X = X} {Y} =
-    right-unit-law-comp-hom-action-Group G X Y
+  action-Group-Large-Precategory =
+    make-Large-Precategory
+      ( action-Group G)
+      ( hom-set-action-Group G)
+      ( λ {l1} {l2} {l3} {X} {Y} {Z} → comp-hom-action-Group G X Y Z)
+      ( λ {l1} {X} → id-hom-action-Group G X)
+      ( λ {l1} {l2} {l3} {l4} {X} {Y} {Z} {W} →
+        associative-comp-hom-action-Group G X Y Z W)
+      ( λ {l1} {l2} {X} {Y} → left-unit-law-comp-hom-action-Group G X Y)
+      ( λ {l1} {l2} {X} {Y} → right-unit-law-comp-hom-action-Group G X Y)
 ```
 
 ### The small precategory of `G`-sets
diff --git a/src/group-theory/precategory-of-groups.lagda.md b/src/group-theory/precategory-of-groups.lagda.md
index 32e64622c1..10f8be5a09 100644
--- a/src/group-theory/precategory-of-groups.lagda.md
+++ b/src/group-theory/precategory-of-groups.lagda.md
@@ -7,37 +7,41 @@ module group-theory.precategory-of-groups where
 <details><summary>Imports</summary>
 
 ```agda
+open import category-theory.full-large-subprecategories
 open import category-theory.large-precategories
+open import category-theory.precategories
 
 open import foundation.universe-levels
 
 open import group-theory.groups
-open import group-theory.homomorphisms-groups
+open import group-theory.precategory-of-semigroups
 ```
 
 </details>
 
 ## Definition
 
+### The precategory of groups as a full subprecategory of the precategory of semigroups
+
+```agda
+Group-Full-Large-Subprecategory :
+  Full-Large-Subprecategory (λ l → l) Semigroup-Large-Precategory
+Group-Full-Large-Subprecategory = is-group-prop-Semigroup
+```
+
+### The large precategory of groups
+
+```agda
+Group-Large-Precategory : Large-Precategory lsuc (_⊔_)
+Group-Large-Precategory =
+  large-precategory-Full-Large-Subprecategory
+    ( Semigroup-Large-Precategory)
+    ( Group-Full-Large-Subprecategory)
+```
+
+### The small precategories of groups
+
 ```agda
-instance
-  Group-Large-Precategory : Large-Precategory lsuc (_⊔_)
-  obj-Large-Precategory Group-Large-Precategory = Group
-  hom-set-Large-Precategory Group-Large-Precategory = hom-set-Group
-  comp-hom-Large-Precategory Group-Large-Precategory {X = G} {H} {K} =
-    comp-hom-Group G H K
-  id-hom-Large-Precategory Group-Large-Precategory {X = G} =
-    id-hom-Group G
-  associative-comp-hom-Large-Precategory
-    Group-Large-Precategory {X = G} {H} {K} {L} =
-    associative-comp-hom-Group G H K L
-  inv-associative-comp-hom-Large-Precategory
-    Group-Large-Precategory {X = G} {H} {K} {L} =
-    inv-associative-comp-hom-Group G H K L
-  left-unit-law-comp-hom-Large-Precategory
-    Group-Large-Precategory {X = G} {H} =
-    left-unit-law-comp-hom-Group G H
-  right-unit-law-comp-hom-Large-Precategory
-    Group-Large-Precategory {X = G} {H} =
-    right-unit-law-comp-hom-Group G H
+Group-Precategory : (l : Level) → Precategory (lsuc l) l
+Group-Precategory = precategory-Large-Precategory Group-Large-Precategory
 ```
diff --git a/src/group-theory/precategory-of-monoids.lagda.md b/src/group-theory/precategory-of-monoids.lagda.md
index e8e90de208..a16650574c 100644
--- a/src/group-theory/precategory-of-monoids.lagda.md
+++ b/src/group-theory/precategory-of-monoids.lagda.md
@@ -8,50 +8,56 @@ module group-theory.precategory-of-monoids where
 
 ```agda
 open import category-theory.large-precategories
+open import category-theory.large-subprecategories
 open import category-theory.precategories
 
+open import foundation.dependent-pair-types
 open import foundation.universe-levels
 
 open import group-theory.homomorphisms-monoids
 open import group-theory.monoids
+open import group-theory.precategory-of-semigroups
 ```
 
 </details>
 
 ## Idea
 
-The **precategory of monoids** consists of monoids and homomorphisms of monoids.
+The {{#concept "precategory of monoids" Agda=Monoid-Large-Precategory}} consists
+of [monoids](group-theory.monoids.md) and
+[homomorphisms of monoids](group-theory.homomorphisms-monoids.md).
 
 ## Definitions
 
+### The precategory of monoids as a subprecategory of the precategory of semigroups
+
+```agda
+Monoid-Large-Subprecategory :
+  Large-Subprecategory (λ l → l) (λ l1 l2 → l2) Semigroup-Large-Precategory
+Monoid-Large-Subprecategory =
+  λ where
+    .subtype-obj-Large-Subprecategory l →
+      is-unital-prop-Semigroup {l}
+    .subtype-hom-Large-Subprecategory G H is-unital-G is-unital-H →
+      preserves-unit-hom-prop-Semigroup (G , is-unital-G) (H , is-unital-H)
+    .contains-id-Large-Subprecategory G is-unital-G →
+      preserves-unit-id-hom-Monoid (G , is-unital-G)
+    .is-closed-under-composition-Large-Subprecategory
+      G H K g f is-unital-G is-unital-H is-unital-K unit-g unit-f →
+      preserves-unit-comp-hom-Monoid
+        ( G , is-unital-G)
+        ( H , is-unital-H)
+        ( K , is-unital-K)
+        ( g , unit-g)
+        ( f , unit-f)
+```
+
 ### The large precategory of monoids
 
 ```agda
-Monoid-Large-Precategory : Large-Precategory lsuc _⊔_
-obj-Large-Precategory
-  Monoid-Large-Precategory =
-  Monoid
-hom-set-Large-Precategory
-  Monoid-Large-Precategory =
-  hom-set-Monoid
-comp-hom-Large-Precategory
-  Monoid-Large-Precategory {X = K} {L} {M} =
-  comp-hom-Monoid K L M
-id-hom-Large-Precategory
-  Monoid-Large-Precategory {X = M} =
-  id-hom-Monoid M
-associative-comp-hom-Large-Precategory
-  Monoid-Large-Precategory {X = K} {L} {M} {N} =
-  associative-comp-hom-Monoid K L M N
-inv-associative-comp-hom-Large-Precategory
-  Monoid-Large-Precategory {X = K} {L} {M} {N} =
-  inv-associative-comp-hom-Monoid K L M N
-left-unit-law-comp-hom-Large-Precategory
-  Monoid-Large-Precategory {X = M} {N} =
-  left-unit-law-comp-hom-Monoid M N
-right-unit-law-comp-hom-Large-Precategory
-  Monoid-Large-Precategory {X = M} {N} =
-  right-unit-law-comp-hom-Monoid M N
+Monoid-Large-Precategory : Large-Precategory lsuc (_⊔_)
+Monoid-Large-Precategory =
+  large-precategory-Large-Subprecategory Monoid-Large-Subprecategory
 ```
 
 ### The precategory of small monoids
diff --git a/src/group-theory/orbits-monoid-actions.lagda.md b/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md
similarity index 89%
rename from src/group-theory/orbits-monoid-actions.lagda.md
rename to src/group-theory/precategory-of-orbits-monoid-actions.lagda.md
index a6766ac596..5a51488a0e 100644
--- a/src/group-theory/orbits-monoid-actions.lagda.md
+++ b/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md
@@ -1,13 +1,12 @@
 # The precategory of orbits of a monoid action
 
 ```agda
-module group-theory.orbits-monoid-actions where
+module group-theory.precategory-of-orbits-monoid-actions where
 ```
 
 <details><summary>Imports</summary>
 
 ```agda
-open import category-theory.composition-operations-on-binary-families-of-sets
 open import category-theory.precategories
 
 open import foundation.contractible-types
@@ -179,19 +178,13 @@ module _
       ( right-unit-law-mul-Monoid M (element-hom-orbit-action-Monoid f))
 
   orbit-monoid-action-Precategory : Precategory l2 (l1 ⊔ l2)
-  pr1 orbit-monoid-action-Precategory = type-action-Monoid M X
-  pr1 (pr2 orbit-monoid-action-Precategory) = hom-orbit-monoid-action-Set
-  pr1 (pr1 (pr2 (pr2 orbit-monoid-action-Precategory))) =
-    comp-hom-orbit-action-Monoid
-  pr2 (pr1 (pr2 (pr2 orbit-monoid-action-Precategory))) =
-    is-associative-witness-associative-composition-operation-binary-family-Set
+  orbit-monoid-action-Precategory =
+    make-Precategory
+      ( type-action-Monoid M X)
       ( hom-orbit-monoid-action-Set)
       ( comp-hom-orbit-action-Monoid)
+      ( id-hom-orbit-action-Monoid)
       ( associative-comp-hom-orbit-action-Monoid)
-  pr1 (pr2 (pr2 (pr2 orbit-monoid-action-Precategory))) =
-    id-hom-orbit-action-Monoid
-  pr1 (pr2 (pr2 (pr2 (pr2 orbit-monoid-action-Precategory)))) =
-    left-unit-law-comp-hom-orbit-action-Monoid
-  pr2 (pr2 (pr2 (pr2 (pr2 orbit-monoid-action-Precategory)))) =
-    right-unit-law-comp-hom-orbit-action-Monoid
+      ( left-unit-law-comp-hom-orbit-action-Monoid)
+      ( right-unit-law-comp-hom-orbit-action-Monoid)
 ```
diff --git a/src/group-theory/precategory-of-semigroups.lagda.md b/src/group-theory/precategory-of-semigroups.lagda.md
index b752aeea3b..264d32e6b7 100644
--- a/src/group-theory/precategory-of-semigroups.lagda.md
+++ b/src/group-theory/precategory-of-semigroups.lagda.md
@@ -23,29 +23,18 @@ Semigroups and semigroup homomorphisms form a precategory.
 
 ## Definition
 
-### The precategory of semigroups
+### The large precategory of semigroups
 
 ```agda
-instance
-  Semigroup-Large-Precategory : Large-Precategory lsuc (_⊔_)
-  obj-Large-Precategory Semigroup-Large-Precategory = Semigroup
-  hom-set-Large-Precategory Semigroup-Large-Precategory = hom-set-Semigroup
-  comp-hom-Large-Precategory Semigroup-Large-Precategory
-    {X = G} {H} {K} =
-    comp-hom-Semigroup G H K
-  id-hom-Large-Precategory Semigroup-Large-Precategory
-    {X = G} =
-    id-hom-Semigroup G
-  associative-comp-hom-Large-Precategory Semigroup-Large-Precategory
-    {X = G} {H} {K} {L} =
-    associative-comp-hom-Semigroup G H K L
-  inv-associative-comp-hom-Large-Precategory Semigroup-Large-Precategory
-    {X = G} {H} {K} {L} =
-    inv-associative-comp-hom-Semigroup G H K L
-  left-unit-law-comp-hom-Large-Precategory Semigroup-Large-Precategory
-    {X = G} {H} =
-    left-unit-law-comp-hom-Semigroup G H
-  right-unit-law-comp-hom-Large-Precategory Semigroup-Large-Precategory
-    {X = G} {H} =
-    right-unit-law-comp-hom-Semigroup G H
+Semigroup-Large-Precategory : Large-Precategory lsuc (_⊔_)
+Semigroup-Large-Precategory =
+  make-Large-Precategory
+    ( Semigroup)
+    ( hom-set-Semigroup)
+    ( λ {l1} {l2} {l3} {G} {H} {K} → comp-hom-Semigroup G H K)
+    ( λ {l} {G} → id-hom-Semigroup G)
+    ( λ {l1} {l2} {l3} {l4} {G} {H} {K} {L} →
+      associative-comp-hom-Semigroup G H K L)
+    ( λ {l1} {l2} {G} {H} → left-unit-law-comp-hom-Semigroup G H)
+    ( λ {l1} {l2} {G} {H} → right-unit-law-comp-hom-Semigroup G H)
 ```
diff --git a/src/group-theory/symmetric-groups.lagda.md b/src/group-theory/symmetric-groups.lagda.md
index 282ff09a2f..bb5b38be9e 100644
--- a/src/group-theory/symmetric-groups.lagda.md
+++ b/src/group-theory/symmetric-groups.lagda.md
@@ -10,12 +10,9 @@ module group-theory.symmetric-groups where
 open import foundation.action-on-identifications-functions
 open import foundation.automorphisms
 open import foundation.dependent-pair-types
-open import foundation.equality-dependent-pair-types
 open import foundation.equivalences
 open import foundation.function-extensionality
-open import foundation.function-types
 open import foundation.identity-types
-open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
 open import foundation.universe-levels
@@ -64,7 +61,9 @@ pr2 (pr2 (is-unital-Semigroup-symmetric-Semigroup X)) = right-unit-law-equiv
 
 is-group-symmetric-Semigroup' :
   {l : Level} (X : Set l) →
-  is-group' (symmetric-Semigroup X) (is-unital-Semigroup-symmetric-Semigroup X)
+  is-group-is-unital-Semigroup
+    ( symmetric-Semigroup X)
+    ( is-unital-Semigroup-symmetric-Semigroup X)
 pr1 (is-group-symmetric-Semigroup' X) = inv-equiv
 pr1 (pr2 (is-group-symmetric-Semigroup' X)) = left-inverse-law-equiv
 pr2 (pr2 (is-group-symmetric-Semigroup' X)) = right-inverse-law-equiv
diff --git a/src/higher-group-theory/homomorphisms-higher-groups.lagda.md b/src/higher-group-theory/homomorphisms-higher-groups.lagda.md
index 72ad473da7..e40ff05191 100644
--- a/src/higher-group-theory/homomorphisms-higher-groups.lagda.md
+++ b/src/higher-group-theory/homomorphisms-higher-groups.lagda.md
@@ -116,12 +116,6 @@ module _
       ( comp-hom-∞-Group G K L h (comp-hom-∞-Group G H K g f))
   associative-comp-hom-∞-Group = associative-comp-pointed-map
 
-  inv-associative-comp-hom-∞-Group :
-    (h : hom-∞-Group K L) (g : hom-∞-Group H K) (f : hom-∞-Group G H) →
-    htpy-hom-∞-Group G L
-      ( comp-hom-∞-Group G K L h (comp-hom-∞-Group G H K g f))
-      ( comp-hom-∞-Group G H L (comp-hom-∞-Group H K L h g) f)
-  inv-associative-comp-hom-∞-Group = inv-associative-comp-pointed-map
 module _
   {l1 l2 : Level} (G : ∞-Group l1) (H : ∞-Group l2)
   where
diff --git a/src/linear-algebra/vectors-on-euclidean-domains.lagda.md b/src/linear-algebra/vectors-on-euclidean-domains.lagda.md
index fae1dfb56d..63019ac835 100644
--- a/src/linear-algebra/vectors-on-euclidean-domains.lagda.md
+++ b/src/linear-algebra/vectors-on-euclidean-domains.lagda.md
@@ -316,7 +316,7 @@ module _
     right-unit-law-add-vec-Euclidean-Domain R
 
   is-group-vec-Euclidean-Domain :
-    (n : ℕ) → is-group (vec-Euclidean-Domain-Semigroup R n)
+    (n : ℕ) → is-group-Semigroup (vec-Euclidean-Domain-Semigroup R n)
   pr1 (is-group-vec-Euclidean-Domain n) = is-unital-vec-Euclidean-Domain n
   pr1 (pr2 (is-group-vec-Euclidean-Domain n)) = neg-vec-Euclidean-Domain
   pr1 (pr2 (pr2 (is-group-vec-Euclidean-Domain n))) =
diff --git a/src/linear-algebra/vectors-on-rings.lagda.md b/src/linear-algebra/vectors-on-rings.lagda.md
index 661b80f987..fbeb4ee829 100644
--- a/src/linear-algebra/vectors-on-rings.lagda.md
+++ b/src/linear-algebra/vectors-on-rings.lagda.md
@@ -265,7 +265,7 @@ module _
   pr1 (pr2 (is-unital-vec-Ring n)) = left-unit-law-add-vec-Ring R
   pr2 (pr2 (is-unital-vec-Ring n)) = right-unit-law-add-vec-Ring R
 
-  is-group-vec-Ring : (n : ℕ) → is-group (vec-Ring-Semigroup R n)
+  is-group-vec-Ring : (n : ℕ) → is-group-Semigroup (vec-Ring-Semigroup R n)
   pr1 (is-group-vec-Ring n) = is-unital-vec-Ring n
   pr1 (pr2 (is-group-vec-Ring n)) = neg-vec-Ring
   pr1 (pr2 (pr2 (is-group-vec-Ring n))) = left-inverse-law-add-vec-Ring
diff --git a/src/order-theory/large-preorders.lagda.md b/src/order-theory/large-preorders.lagda.md
index adecba1d79..814b16dabb 100644
--- a/src/order-theory/large-preorders.lagda.md
+++ b/src/order-theory/large-preorders.lagda.md
@@ -14,6 +14,7 @@ open import foundation.identity-types
 open import foundation.large-binary-relations
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.universe-levels
 
 open import order-theory.preorders
@@ -142,12 +143,10 @@ module _
     transitive-leq-Large-Preorder P x y z
   id-hom-Large-Precategory large-precategory-Large-Preorder {X = x} =
     refl-leq-Large-Preorder P x
-  associative-comp-hom-Large-Precategory large-precategory-Large-Preorder
+  involutive-eq-associative-comp-hom-Large-Precategory
+    large-precategory-Large-Preorder
     {X = x} {W = w} h g f =
-    eq-is-prop (is-prop-leq-Large-Preorder P x w)
-  inv-associative-comp-hom-Large-Precategory large-precategory-Large-Preorder
-    {X = x} {W = w} h g f =
-    eq-is-prop (is-prop-leq-Large-Preorder P x w)
+    involutive-eq-eq (eq-is-prop (is-prop-leq-Large-Preorder P x w))
   left-unit-law-comp-hom-Large-Precategory large-precategory-Large-Preorder
     {X = x} {y} f =
     eq-is-prop (is-prop-leq-Large-Preorder P x y)
diff --git a/src/order-theory/order-preserving-maps-large-posets.lagda.md b/src/order-theory/order-preserving-maps-large-posets.lagda.md
index 6c8ff099cc..f4cd9eb033 100644
--- a/src/order-theory/order-preserving-maps-large-posets.lagda.md
+++ b/src/order-theory/order-preserving-maps-large-posets.lagda.md
@@ -9,9 +9,6 @@ module order-theory.order-preserving-maps-large-posets where
 ```agda
 open import foundation.universe-levels
 
-open import foundation-core.function-types
-open import foundation-core.homotopies
-
 open import order-theory.large-posets
 open import order-theory.order-preserving-maps-large-preorders
 open import order-theory.similarity-of-elements-large-posets
@@ -192,20 +189,6 @@ module _
       ( h)
       ( g)
       ( f)
-
-  inv-associative-comp-hom-Large-Poset :
-    htpy-hom-Large-Poset P S
-      ( comp-hom-Large-Poset P R S h (comp-hom-Large-Poset P Q R g f))
-      ( comp-hom-Large-Poset P Q S (comp-hom-Large-Poset Q R S h g) f)
-  inv-associative-comp-hom-Large-Poset =
-    inv-associative-comp-hom-Large-Preorder
-      ( large-preorder-Large-Poset P)
-      ( large-preorder-Large-Poset Q)
-      ( large-preorder-Large-Poset R)
-      ( large-preorder-Large-Poset S)
-      ( h)
-      ( g)
-      ( f)
 ```
 
 ### Composition of order preserving maps satisfies left and right unit laws
diff --git a/src/order-theory/order-preserving-maps-large-preorders.lagda.md b/src/order-theory/order-preserving-maps-large-preorders.lagda.md
index bf603766c6..bca9ee11b2 100644
--- a/src/order-theory/order-preserving-maps-large-preorders.lagda.md
+++ b/src/order-theory/order-preserving-maps-large-preorders.lagda.md
@@ -159,12 +159,6 @@ module _
       ( comp-hom-Large-Preorder P Q S (comp-hom-Large-Preorder Q R S h g) f)
       ( comp-hom-Large-Preorder P R S h (comp-hom-Large-Preorder P Q R g f))
   associative-comp-hom-Large-Preorder = refl-htpy
-
-  inv-associative-comp-hom-Large-Preorder :
-    htpy-hom-Large-Preorder P S
-      ( comp-hom-Large-Preorder P R S h (comp-hom-Large-Preorder P Q R g f))
-      ( comp-hom-Large-Preorder P Q S (comp-hom-Large-Preorder Q R S h g) f)
-  inv-associative-comp-hom-Large-Preorder = refl-htpy
 ```
 
 ### Composition of order preserving maps satisfies left and right unit laws
diff --git a/src/order-theory/order-preserving-maps-posets.lagda.md b/src/order-theory/order-preserving-maps-posets.lagda.md
index 3c5f3ef61e..cfb1e1f651 100644
--- a/src/order-theory/order-preserving-maps-posets.lagda.md
+++ b/src/order-theory/order-preserving-maps-posets.lagda.md
@@ -12,6 +12,7 @@ open import foundation.function-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.sets
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtypes
 open import foundation.torsorial-type-families
 open import foundation.universe-levels
@@ -197,12 +198,12 @@ module _
       ( preorder-Poset R)
       ( preorder-Poset S)
 
-  inv-associative-comp-hom-Poset :
+  involutive-eq-associative-comp-hom-Poset :
     (h : hom-Poset R S) (g : hom-Poset Q R) (f : hom-Poset P Q) →
-    comp-hom-Poset P R S h (comp-hom-Poset P Q R g f) ＝
-    comp-hom-Poset P Q S (comp-hom-Poset Q R S h g) f
-  inv-associative-comp-hom-Poset =
-    inv-associative-comp-hom-Preorder
+    comp-hom-Poset P Q S (comp-hom-Poset Q R S h g) f ＝ⁱ
+    comp-hom-Poset P R S h (comp-hom-Poset P Q R g f)
+  involutive-eq-associative-comp-hom-Poset =
+    involutive-eq-associative-comp-hom-Preorder
       ( preorder-Poset P)
       ( preorder-Poset Q)
       ( preorder-Poset R)
diff --git a/src/order-theory/order-preserving-maps-preorders.lagda.md b/src/order-theory/order-preserving-maps-preorders.lagda.md
index 8021299f9d..b3bbc7d62b 100644
--- a/src/order-theory/order-preserving-maps-preorders.lagda.md
+++ b/src/order-theory/order-preserving-maps-preorders.lagda.md
@@ -15,6 +15,7 @@ open import foundation.homotopies
 open import foundation.homotopy-induction
 open import foundation.identity-types
 open import foundation.propositions
+open import foundation.strictly-involutive-identity-types
 open import foundation.subtype-identity-principle
 open import foundation.torsorial-type-families
 open import foundation.universe-levels
@@ -209,12 +210,9 @@ module _
       ( comp-hom-Preorder P R S h (comp-hom-Preorder P Q R g f))
       ( refl-htpy)
 
-  inv-associative-comp-hom-Preorder :
-    comp-hom-Preorder P R S h (comp-hom-Preorder P Q R g f) ＝
-    comp-hom-Preorder P Q S (comp-hom-Preorder Q R S h g) f
-  inv-associative-comp-hom-Preorder =
-    eq-htpy-hom-Preorder P S
-      ( comp-hom-Preorder P R S h (comp-hom-Preorder P Q R g f))
-      ( comp-hom-Preorder P Q S (comp-hom-Preorder Q R S h g) f)
-      ( refl-htpy)
+  involutive-eq-associative-comp-hom-Preorder :
+    comp-hom-Preorder P Q S (comp-hom-Preorder Q R S h g) f ＝ⁱ
+    comp-hom-Preorder P R S h (comp-hom-Preorder P Q R g f)
+  involutive-eq-associative-comp-hom-Preorder =
+    involutive-eq-eq associative-comp-hom-Preorder
 ```
diff --git a/src/order-theory/precategory-of-posets.lagda.md b/src/order-theory/precategory-of-posets.lagda.md
index 6eee88f969..29876db4bc 100644
--- a/src/order-theory/precategory-of-posets.lagda.md
+++ b/src/order-theory/precategory-of-posets.lagda.md
@@ -20,8 +20,8 @@ open import order-theory.posets
 
 ## Idea
 
-The **(large) precategory of posets** consists of
-[posets](order-theory.posets.md) and
+The {{#concept "(large) precategory of posets" Agda=Poset-Large-Precategory}}
+consists of [posets](order-theory.posets.md) and
 [order preserving maps](order-theory.order-preserving-maps-posets.md).
 
 ## Definitions
@@ -36,14 +36,16 @@ parametric-Poset-Large-Precategory :
     ( λ l1 l2 → α l1 ⊔ β l1 ⊔ α l2 ⊔ β l2)
 parametric-Poset-Large-Precategory α β =
   λ where
-    .obj-Large-Precategory l → Poset (α l) (β l)
-    .hom-set-Large-Precategory → hom-set-Poset
-    .comp-hom-Large-Precategory {X = X} {Y} {Z} → comp-hom-Poset X Y Z
-    .id-hom-Large-Precategory {X = X} → id-hom-Poset X
-    .associative-comp-hom-Large-Precategory {X = X} {Y} {Z} {W} →
-      associative-comp-hom-Poset X Y Z W
-    .inv-associative-comp-hom-Large-Precategory {X = X} {Y} {Z} {W} →
-      inv-associative-comp-hom-Poset X Y Z W
+    .obj-Large-Precategory l →
+      Poset (α l) (β l)
+    .hom-set-Large-Precategory →
+      hom-set-Poset
+    .comp-hom-Large-Precategory {X = X} {Y} {Z} →
+      comp-hom-Poset X Y Z
+    .id-hom-Large-Precategory {X = X} →
+      id-hom-Poset X
+    .involutive-eq-associative-comp-hom-Large-Precategory {X = X} {Y} {Z} {W} →
+      involutive-eq-associative-comp-hom-Poset X Y Z W
     .left-unit-law-comp-hom-Large-Precategory {X = X} {Y} →
       left-unit-law-comp-hom-Poset X Y
     .right-unit-law-comp-hom-Large-Precategory {X = X} {Y} →
diff --git a/src/order-theory/preorders.lagda.md b/src/order-theory/preorders.lagda.md
index 6ad7c5df71..2bc2aa9ba2 100644
--- a/src/order-theory/preorders.lagda.md
+++ b/src/order-theory/preorders.lagda.md
@@ -137,19 +137,16 @@ module _
   where
 
   precategory-Preorder : Precategory l1 l2
-  pr1 precategory-Preorder = type-Preorder X
-  pr1 (pr2 precategory-Preorder) x y = set-Prop (leq-Preorder-Prop X x y)
-  pr1 (pr1 (pr2 (pr2 precategory-Preorder))) {x} {y} {z} =
-    is-transitive-leq-Preorder X x y z
-  pr1 (pr2 (pr1 (pr2 (pr2 precategory-Preorder))) {x} {y} {z} {w} h g f) =
-    eq-is-prop (is-prop-type-Prop (leq-Preorder-Prop X x w))
-  pr2 (pr2 (pr1 (pr2 (pr2 precategory-Preorder))) {x} {y} {z} {w} h g f) =
-    eq-is-prop (is-prop-type-Prop (leq-Preorder-Prop X x w))
-  pr1 (pr2 (pr2 (pr2 precategory-Preorder))) = refl-leq-Preorder X
-  pr1 (pr2 (pr2 (pr2 (pr2 precategory-Preorder)))) {x} {y} f =
-    eq-is-prop (is-prop-type-Prop (leq-Preorder-Prop X x y))
-  pr2 (pr2 (pr2 (pr2 (pr2 precategory-Preorder)))) {x} {y} f =
-    eq-is-prop (is-prop-type-Prop (leq-Preorder-Prop X x y))
+  precategory-Preorder =
+    make-Precategory
+      ( type-Preorder X)
+      ( λ x y → set-Prop (leq-Preorder-Prop X x y))
+      ( λ {x} {y} {z} → is-transitive-leq-Preorder X x y z)
+      ( refl-leq-Preorder X)
+      ( λ {x} {y} {z} {w} h g f →
+        eq-is-prop (is-prop-type-Prop (leq-Preorder-Prop X x w)))
+      ( λ {x} {y} f → eq-is-prop (is-prop-type-Prop (leq-Preorder-Prop X x y)))
+      ( λ {x} {y} f → eq-is-prop (is-prop-type-Prop (leq-Preorder-Prop X x y)))
 
 module _
   {l1 l2 : Level} (C : Precategory l1 l2)
diff --git a/src/ring-theory/category-of-cyclic-rings.lagda.md b/src/ring-theory/category-of-cyclic-rings.lagda.md
index b513d8c8d1..27c4f61a07 100644
--- a/src/ring-theory/category-of-cyclic-rings.lagda.md
+++ b/src/ring-theory/category-of-cyclic-rings.lagda.md
@@ -8,27 +8,27 @@ module ring-theory.category-of-cyclic-rings where
 
 ```agda
 open import category-theory.categories
-open import category-theory.isomorphisms-in-large-precategories
+open import category-theory.full-large-subprecategories
 open import category-theory.large-categories
 open import category-theory.large-precategories
 
-open import foundation.fundamental-theorem-of-identity-types
-open import foundation.subtype-identity-principle
 open import foundation.universe-levels
 
 open import order-theory.large-posets
 
+open import ring-theory.category-of-rings
 open import ring-theory.cyclic-rings
 open import ring-theory.homomorphisms-cyclic-rings
-open import ring-theory.isomorphisms-rings
+open import ring-theory.precategory-of-rings
 ```
 
 </details>
 
 ## Idea
 
-The **large category of cyclic rings** is the
-[large category](category-theory.large-categories.md) consisting of
+The
+{{#concept "large category of cyclic rings" Agda=Cyclic-Ring-Large-Category}} is
+the [large category](category-theory.large-categories.md) consisting of
 [cyclic rings](ring-theory.cyclic-rings.md) and
 [ring homomorphisms](ring-theory.homomorphisms-cyclic-rings.md).
 
@@ -40,34 +40,22 @@ rings is constructed in the file
 
 ## Definition
 
+### The precategory of cyclic rings as a full subprecategory of the precategory of rings
+
+```agda
+Cyclic-Ring-Full-Large-Subprecategory :
+  Full-Large-Subprecategory (λ l → l) Ring-Large-Precategory
+Cyclic-Ring-Full-Large-Subprecategory = is-cyclic-prop-Ring
+```
+
 ### The large precategory of cyclic rings
 
 ```agda
-Cyclic-Ring-Large-Precategory : Large-Precategory lsuc _⊔_
-obj-Large-Precategory
-  Cyclic-Ring-Large-Precategory =
-  Cyclic-Ring
-hom-set-Large-Precategory
-  Cyclic-Ring-Large-Precategory =
-  hom-set-Cyclic-Ring
-comp-hom-Large-Precategory
-  Cyclic-Ring-Large-Precategory {X = R} {Y = S} {Z = T} =
-  comp-hom-Cyclic-Ring R S T
-id-hom-Large-Precategory
-  Cyclic-Ring-Large-Precategory {X = R} =
-  id-hom-Cyclic-Ring R
-associative-comp-hom-Large-Precategory
-  Cyclic-Ring-Large-Precategory {X = R} {Y = S} {Z = T} {W = U} =
-  associative-comp-hom-Cyclic-Ring R S T U
-inv-associative-comp-hom-Large-Precategory
-  Cyclic-Ring-Large-Precategory {X = R} {Y = S} {Z = T} {W = U} =
-  inv-associative-comp-hom-Cyclic-Ring R S T U
-left-unit-law-comp-hom-Large-Precategory
-  Cyclic-Ring-Large-Precategory {X = R} {Y = S} =
-  left-unit-law-comp-hom-Cyclic-Ring R S
-right-unit-law-comp-hom-Large-Precategory
-  Cyclic-Ring-Large-Precategory {X = R} {Y = S} =
-  right-unit-law-comp-hom-Cyclic-Ring R S
+Cyclic-Ring-Large-Precategory : Large-Precategory lsuc (_⊔_)
+Cyclic-Ring-Large-Precategory =
+  large-precategory-Full-Large-Subprecategory
+    ( Ring-Large-Precategory)
+    ( Cyclic-Ring-Full-Large-Subprecategory)
 ```
 
 ### The large category of cyclic rings
@@ -76,17 +64,13 @@ right-unit-law-comp-hom-Large-Precategory
 abstract
   is-large-category-Cyclic-Ring-Large-Category :
     is-large-category-Large-Precategory Cyclic-Ring-Large-Precategory
-  is-large-category-Cyclic-Ring-Large-Category R =
-    fundamental-theorem-id
-      ( is-torsorial-Eq-subtype
-        ( is-torsorial-iso-Ring (ring-Cyclic-Ring R))
-        ( is-prop-is-cyclic-Ring)
-        ( ring-Cyclic-Ring R)
-        ( id-iso-Ring (ring-Cyclic-Ring R))
-        ( is-cyclic-Cyclic-Ring R))
-      ( iso-eq-Large-Precategory Cyclic-Ring-Large-Precategory R)
-
-Cyclic-Ring-Large-Category : Large-Category lsuc _⊔_
+  is-large-category-Cyclic-Ring-Large-Category =
+    is-large-category-large-precategory-is-large-category-Full-Large-Subprecategory
+      ( Ring-Large-Precategory)
+      ( Cyclic-Ring-Full-Large-Subprecategory)
+      ( is-large-category-Ring-Large-Category)
+
+Cyclic-Ring-Large-Category : Large-Category lsuc (_⊔_)
 large-precategory-Large-Category
   Cyclic-Ring-Large-Category =
   Cyclic-Ring-Large-Precategory
diff --git a/src/ring-theory/category-of-rings.lagda.md b/src/ring-theory/category-of-rings.lagda.md
index 864f8110f8..bd3bfa299d 100644
--- a/src/ring-theory/category-of-rings.lagda.md
+++ b/src/ring-theory/category-of-rings.lagda.md
@@ -34,7 +34,7 @@ is-large-category-Ring-Large-Category :
 is-large-category-Ring-Large-Category =
   is-equiv-iso-eq-Ring
 
-Ring-Large-Category : Large-Category lsuc _⊔_
+Ring-Large-Category : Large-Category lsuc (_⊔_)
 large-precategory-Large-Category Ring-Large-Category =
   Ring-Large-Precategory
 is-large-category-Large-Category Ring-Large-Category =
diff --git a/src/ring-theory/groups-of-units-rings.lagda.md b/src/ring-theory/groups-of-units-rings.lagda.md
index da0967105b..16e293804e 100644
--- a/src/ring-theory/groups-of-units-rings.lagda.md
+++ b/src/ring-theory/groups-of-units-rings.lagda.md
@@ -117,11 +117,14 @@ module _
     right-inverse-law-mul-core-Monoid (multiplicative-monoid-Ring R)
 
   is-group-group-of-units-Ring' :
-    is-group' semigroup-group-of-units-Ring is-unital-group-of-units-Ring
+    is-group-is-unital-Semigroup
+      ( semigroup-group-of-units-Ring)
+      ( is-unital-group-of-units-Ring)
   is-group-group-of-units-Ring' =
     is-group-core-Monoid' (multiplicative-monoid-Ring R)
 
-  is-group-group-of-units-Ring : is-group semigroup-group-of-units-Ring
+  is-group-group-of-units-Ring :
+    is-group-Semigroup semigroup-group-of-units-Ring
   is-group-group-of-units-Ring =
     is-group-core-Monoid (multiplicative-monoid-Ring R)
 
diff --git a/src/ring-theory/homomorphisms-cyclic-rings.lagda.md b/src/ring-theory/homomorphisms-cyclic-rings.lagda.md
index 37cdae96bf..8bdb6d76cf 100644
--- a/src/ring-theory/homomorphisms-cyclic-rings.lagda.md
+++ b/src/ring-theory/homomorphisms-cyclic-rings.lagda.md
@@ -204,19 +204,6 @@ module _
       ( ring-Cyclic-Ring S)
       ( ring-Cyclic-Ring T)
       ( ring-Cyclic-Ring U)
-
-  inv-associative-comp-hom-Cyclic-Ring :
-    (h : hom-Cyclic-Ring T U)
-    (g : hom-Cyclic-Ring S T)
-    (f : hom-Cyclic-Ring R S) →
-    comp-hom-Cyclic-Ring R T U h (comp-hom-Cyclic-Ring R S T g f) ＝
-    comp-hom-Cyclic-Ring R S U (comp-hom-Cyclic-Ring S T U h g) f
-  inv-associative-comp-hom-Cyclic-Ring =
-    inv-associative-comp-hom-Ring
-      ( ring-Cyclic-Ring R)
-      ( ring-Cyclic-Ring S)
-      ( ring-Cyclic-Ring T)
-      ( ring-Cyclic-Ring U)
 ```
 
 ## See also
diff --git a/src/ring-theory/homomorphisms-rings.lagda.md b/src/ring-theory/homomorphisms-rings.lagda.md
index 9fb1a19db9..95f466ea7c 100644
--- a/src/ring-theory/homomorphisms-rings.lagda.md
+++ b/src/ring-theory/homomorphisms-rings.lagda.md
@@ -355,15 +355,6 @@ module _
       ( comp-hom-Ring R S U (comp-hom-Ring S T U h g) f)
       ( comp-hom-Ring R T U h (comp-hom-Ring R S T g f))
       ( refl-htpy)
-
-  inv-associative-comp-hom-Ring :
-    comp-hom-Ring R T U h (comp-hom-Ring R S T g f) ＝
-    comp-hom-Ring R S U (comp-hom-Ring S T U h g) f
-  inv-associative-comp-hom-Ring =
-    eq-htpy-hom-Ring R U
-      ( comp-hom-Ring R T U h (comp-hom-Ring R S T g f))
-      ( comp-hom-Ring R S U (comp-hom-Ring S T U h g) f)
-      ( refl-htpy)
 ```
 
 ### Unit laws for composition of ring homomorphisms
diff --git a/src/ring-theory/homomorphisms-semirings.lagda.md b/src/ring-theory/homomorphisms-semirings.lagda.md
index 7f709a2561..9bd4acad47 100644
--- a/src/ring-theory/homomorphisms-semirings.lagda.md
+++ b/src/ring-theory/homomorphisms-semirings.lagda.md
@@ -335,15 +335,6 @@ module _
       ( comp-hom-Semiring R S U (comp-hom-Semiring S T U h g) f)
       ( comp-hom-Semiring R T U h (comp-hom-Semiring R S T g f))
       ( refl-htpy)
-
-  inv-associative-comp-hom-Semiring :
-    comp-hom-Semiring R T U h (comp-hom-Semiring R S T g f) ＝
-    comp-hom-Semiring R S U (comp-hom-Semiring S T U h g) f
-  inv-associative-comp-hom-Semiring =
-    eq-htpy-hom-Semiring R U
-      ( comp-hom-Semiring R T U h (comp-hom-Semiring R S T g f))
-      ( comp-hom-Semiring R S U (comp-hom-Semiring S T U h g) f)
-      ( refl-htpy)
 ```
 
 ### Unit laws for composition of homomorphisms of semirings
diff --git a/src/ring-theory/poset-of-cyclic-rings.lagda.md b/src/ring-theory/poset-of-cyclic-rings.lagda.md
index 0ec90ccaeb..ef3023bcd0 100644
--- a/src/ring-theory/poset-of-cyclic-rings.lagda.md
+++ b/src/ring-theory/poset-of-cyclic-rings.lagda.md
@@ -31,7 +31,7 @@ The large poset of cyclic rings is dual to the large poset of
 ### The large poset of cyclic rings
 
 ```agda
-Cyclic-Ring-Large-Poset : Large-Poset lsuc _⊔_
+Cyclic-Ring-Large-Poset : Large-Poset lsuc (_⊔_)
 Cyclic-Ring-Large-Poset =
   large-poset-Large-Category
     ( Cyclic-Ring-Large-Category)
diff --git a/src/ring-theory/precategory-of-rings.lagda.md b/src/ring-theory/precategory-of-rings.lagda.md
index fceb7d32a2..e4d937c4e3 100644
--- a/src/ring-theory/precategory-of-rings.lagda.md
+++ b/src/ring-theory/precategory-of-rings.lagda.md
@@ -28,31 +28,16 @@ and [ring homomorphisms](ring-theory.homomorphisms-rings.md).
 ### The large precategory of rings
 
 ```agda
-Ring-Large-Precategory : Large-Precategory lsuc _⊔_
-obj-Large-Precategory
-  Ring-Large-Precategory =
-  Ring
-hom-set-Large-Precategory
-  Ring-Large-Precategory =
-  hom-set-Ring
-comp-hom-Large-Precategory
-  Ring-Large-Precategory {X = R} {S} {T} =
-  comp-hom-Ring R S T
-id-hom-Large-Precategory
-  Ring-Large-Precategory {X = R} =
-  id-hom-Ring R
-associative-comp-hom-Large-Precategory
-  Ring-Large-Precategory {X = R} {S} {T} {U} =
-  associative-comp-hom-Ring R S T U
-inv-associative-comp-hom-Large-Precategory
-  Ring-Large-Precategory {X = R} {S} {T} {U} =
-  inv-associative-comp-hom-Ring R S T U
-left-unit-law-comp-hom-Large-Precategory
-  Ring-Large-Precategory {X = R} {S} =
-  left-unit-law-comp-hom-Ring R S
-right-unit-law-comp-hom-Large-Precategory
-  Ring-Large-Precategory {X = R} {S} =
-  right-unit-law-comp-hom-Ring R S
+Ring-Large-Precategory : Large-Precategory (lsuc) (_⊔_)
+Ring-Large-Precategory =
+  make-Large-Precategory
+    ( Ring)
+    ( hom-set-Ring)
+    ( λ {l1} {l2} {l3} {R} {S} {T} → comp-hom-Ring R S T)
+    ( λ {l} {R} → id-hom-Ring R)
+    ( λ {l1} {l2} {l3} {l4} {R} {S} {T} {U} → associative-comp-hom-Ring R S T U)
+    ( λ {l1} {l2} {R} {S} → left-unit-law-comp-hom-Ring R S)
+    ( λ {l1} {l2} {R} {S} → right-unit-law-comp-hom-Ring R S)
 ```
 
 ### The precategory or rings of universe level `l`
diff --git a/src/ring-theory/precategory-of-semirings.lagda.md b/src/ring-theory/precategory-of-semirings.lagda.md
index a05398d556..8b84a0d3e0 100644
--- a/src/ring-theory/precategory-of-semirings.lagda.md
+++ b/src/ring-theory/precategory-of-semirings.lagda.md
@@ -10,7 +10,6 @@ module ring-theory.precategory-of-semirings where
 open import category-theory.large-precategories
 open import category-theory.precategories
 
-open import foundation.identity-types
 open import foundation.universe-levels
 
 open import ring-theory.homomorphisms-semirings
@@ -21,42 +20,29 @@ open import ring-theory.semirings
 
 ## Idea
 
-The **precategory of semirings** consists of semirings and homomorphisms of
-semirings.
+The {{#concet "precategory of semirings" Agda=Semiring-Large-Precategory}}
+consists of [semirings](ring-theory.semirings.md) and
+[homomorphisms of semirings](ring-theory.homomorphisms-semirings.md).
 
 ## Definitions
 
 ### The large precategory of semirings
 
 ```agda
-Semiring-Large-Precategory : Large-Precategory lsuc _⊔_
-obj-Large-Precategory
-  Semiring-Large-Precategory =
-  Semiring
-hom-set-Large-Precategory
-  Semiring-Large-Precategory =
-  hom-set-Semiring
-comp-hom-Large-Precategory
-  Semiring-Large-Precategory {X = R} {S} {T} =
-  comp-hom-Semiring R S T
-id-hom-Large-Precategory
-  Semiring-Large-Precategory {X = R} =
-  id-hom-Semiring R
-associative-comp-hom-Large-Precategory
-  Semiring-Large-Precategory {X = R} {S} {T} {U} =
-  associative-comp-hom-Semiring R S T U
-inv-associative-comp-hom-Large-Precategory
-  Semiring-Large-Precategory {X = R} {S} {T} {U} =
-  inv-associative-comp-hom-Semiring R S T U
-left-unit-law-comp-hom-Large-Precategory
-  Semiring-Large-Precategory {X = R} {S} =
-  left-unit-law-comp-hom-Semiring R S
-right-unit-law-comp-hom-Large-Precategory
-  Semiring-Large-Precategory {X = R} {S} =
-  right-unit-law-comp-hom-Semiring R S
+Semiring-Large-Precategory : Large-Precategory lsuc (_⊔_)
+Semiring-Large-Precategory =
+  make-Large-Precategory
+    ( Semiring)
+    ( hom-set-Semiring)
+    ( λ {l1} {l2} {l3} {R} {S} {T} → comp-hom-Semiring R S T)
+    ( λ {l} {R} → id-hom-Semiring R)
+    ( λ {l1} {l2} {l3} {l4} {R} {S} {T} {U} →
+      associative-comp-hom-Semiring R S T U)
+    ( λ {l1} {l2} {R} {S} → left-unit-law-comp-hom-Semiring R S)
+    ( λ {l1} {l2} {R} {S} → right-unit-law-comp-hom-Semiring R S)
 ```
 
-### The precategory of semirings of universe level `l`
+### The precategory of small semirings
 
 ```agda
 Semiring-Precategory : (l : Level) → Precategory (lsuc l) l
diff --git a/src/ring-theory/rings.lagda.md b/src/ring-theory/rings.lagda.md
index 69bdbd6ffa..757f7994ee 100644
--- a/src/ring-theory/rings.lagda.md
+++ b/src/ring-theory/rings.lagda.md
@@ -123,7 +123,8 @@ module _
     Id (add-Ring (add-Ring x y) z) (add-Ring x (add-Ring y z))
   associative-add-Ring = associative-add-Ab (ab-Ring R)
 
-  is-group-additive-semigroup-Ring : is-group (additive-semigroup-Ring R)
+  is-group-additive-semigroup-Ring :
+    is-group-Semigroup (additive-semigroup-Ring R)
   is-group-additive-semigroup-Ring = is-group-Ab (ab-Ring R)
 
   commutative-add-Ring : (x y : type-Ring R) → Id (add-Ring x y) (add-Ring y x)
@@ -289,7 +290,8 @@ module _
   {l : Level} (R : Ring l)
   where
 
-  has-negatives-Ring : is-group' (additive-semigroup-Ring R) (has-zero-Ring R)
+  has-negatives-Ring :
+    is-group-is-unital-Semigroup (additive-semigroup-Ring R) (has-zero-Ring R)
   has-negatives-Ring = has-negatives-Ab (ab-Ring R)
 
   neg-Ring : type-Ring R → type-Ring R
diff --git a/src/ring-theory/semirings.lagda.md b/src/ring-theory/semirings.lagda.md
index 66c1cbe91c..982869d8d3 100644
--- a/src/ring-theory/semirings.lagda.md
+++ b/src/ring-theory/semirings.lagda.md
@@ -30,11 +30,13 @@ open import group-theory.semigroups
 
 ## Idea
 
-The concept of semiring vastly generalizes the arithmetical structure on the
-natural numbers. A semiring consists of a set equipped with addition and
-multiplication, where the addition operation gives the ring the structure of a
-commutative monoid, and the multiplication is associative, unital, and
-distributive over addition.
+The concept of a _semiring_ vastly generalizes the arithmetical structure on the
+[natural numbers](elementary-number-theory.natural-numbers.md). A
+{{#concept "semiring" Agda=Semiring}} consists of a
+[set](foundation-core.sets.md) equipped with addition and multiplication, where
+the addition operation gives the semiring the structure of a
+[commutative monoid](group-theory.commutative-monoids.md), and the
+multiplication is associative, unital, and distributive over addition.
 
 ## Definitions
 
diff --git a/src/species/morphisms-finite-species.lagda.md b/src/species/morphisms-finite-species.lagda.md
index bfb389c0c0..6feb38556a 100644
--- a/src/species/morphisms-finite-species.lagda.md
+++ b/src/species/morphisms-finite-species.lagda.md
@@ -91,12 +91,6 @@ module _
     comp-hom-species-𝔽 F G K (comp-hom-species-𝔽 G H K h g) f ＝
     comp-hom-species-𝔽 F H K h (comp-hom-species-𝔽 F G H g f)
   associative-comp-hom-species-𝔽 h g f = refl
-
-  inv-associative-comp-hom-species-𝔽 :
-    (h : hom-species-𝔽 H K) (g : hom-species-𝔽 G H) (f : hom-species-𝔽 F G) →
-    comp-hom-species-𝔽 F H K h (comp-hom-species-𝔽 F G H g f) ＝
-    comp-hom-species-𝔽 F G K (comp-hom-species-𝔽 G H K h g) f
-  inv-associative-comp-hom-species-𝔽 h g f = refl
 ```
 
 ### The unit laws for composition of homomorphisms of finite species
diff --git a/src/species/morphisms-species-of-types.lagda.md b/src/species/morphisms-species-of-types.lagda.md
index 5773cfc7c3..713b51777c 100644
--- a/src/species/morphisms-species-of-types.lagda.md
+++ b/src/species/morphisms-species-of-types.lagda.md
@@ -114,11 +114,6 @@ module _
     comp-hom-species-types (comp-hom-species-types h g) f ＝
     comp-hom-species-types h (comp-hom-species-types g f)
   associative-comp-hom-species-types = refl
-
-  inv-associative-comp-hom-species-types :
-    comp-hom-species-types h (comp-hom-species-types g f) ＝
-    comp-hom-species-types (comp-hom-species-types h g) f
-  inv-associative-comp-hom-species-types = refl
 ```
 
 ### Unit laws of composition
diff --git a/src/species/precategory-of-finite-species.lagda.md b/src/species/precategory-of-finite-species.lagda.md
index 6f78d29f80..a5cc4ed1e4 100644
--- a/src/species/precategory-of-finite-species.lagda.md
+++ b/src/species/precategory-of-finite-species.lagda.md
@@ -19,31 +19,25 @@ open import species.species-of-finite-types
 
 ## Idea
 
-The **precategory of finite species** consists of finite species and
-homomorphisms of finite species.
+The
+{{#concept "precategory of finite species" Agda=species-𝔽-Large-Precategory}}
+consists of [finite species](species.species-of-finite-types.md) and
+[homomorphisms of finite species](species.morphisms-finite-species.md).
 
 ## Definition
 
 ```agda
 species-𝔽-Large-Precategory :
-  (l1 : Level) →
-  Large-Precategory (λ l → lsuc l1 ⊔ lsuc l) (λ l2 l3 → lsuc l1 ⊔ l2 ⊔ l3)
-obj-Large-Precategory (species-𝔽-Large-Precategory l1) = species-𝔽 l1
-hom-set-Large-Precategory (species-𝔽-Large-Precategory l1) = hom-set-species-𝔽
-comp-hom-Large-Precategory (species-𝔽-Large-Precategory l1) {X = F} {G} {H} =
-  comp-hom-species-𝔽 F G H
-id-hom-Large-Precategory (species-𝔽-Large-Precategory l1) {X = F} =
-  id-hom-species-𝔽 F
-associative-comp-hom-Large-Precategory
-  ( species-𝔽-Large-Precategory l1) {X = F} {G} {H} {K} h g f =
-  associative-comp-hom-species-𝔽 F G H K h g f
-inv-associative-comp-hom-Large-Precategory
-  ( species-𝔽-Large-Precategory l1) {X = F} {G} {H} {K} h g f =
-  inv-associative-comp-hom-species-𝔽 F G H K h g f
-left-unit-law-comp-hom-Large-Precategory
-  ( species-𝔽-Large-Precategory l1) {X = F} {G} f =
-  left-unit-law-comp-hom-species-𝔽 F G f
-right-unit-law-comp-hom-Large-Precategory
-  ( species-𝔽-Large-Precategory l1) {X = F} {G} f =
-  right-unit-law-comp-hom-species-𝔽 F G f
+  (l : Level) →
+  Large-Precategory (λ l1 → lsuc l ⊔ lsuc l1) (λ l2 l3 → lsuc l ⊔ l2 ⊔ l3)
+species-𝔽-Large-Precategory l =
+  make-Large-Precategory
+    ( species-𝔽 l)
+    ( hom-set-species-𝔽)
+    ( λ {l1} {l2} {l3} {F} {G} {H} → comp-hom-species-𝔽 F G H)
+    ( λ {l1} {F} → id-hom-species-𝔽 F)
+    ( λ {l1} {l2} {l3} {l4} {F} {G} {H} {K} →
+      associative-comp-hom-species-𝔽 F G H K)
+    ( λ {l1} {l2} {F} {G} → left-unit-law-comp-hom-species-𝔽 F G)
+    ( λ {l1} {l2} {F} {G} → right-unit-law-comp-hom-species-𝔽 F G)
 ```
diff --git a/src/trees/undirected-trees.lagda.md b/src/trees/undirected-trees.lagda.md
index f11286e826..f60cce11cf 100644
--- a/src/trees/undirected-trees.lagda.md
+++ b/src/trees/undirected-trees.lagda.md
@@ -1,4 +1,4 @@
-# Undirected rees
+# Undirected trees
 
 ```agda
 module trees.undirected-trees where
diff --git a/src/universal-algebra/algebraic-theory-of-groups.lagda.md b/src/universal-algebra/algebraic-theory-of-groups.lagda.md
index 5645a87ba0..719d008255 100644
--- a/src/universal-algebra/algebraic-theory-of-groups.lagda.md
+++ b/src/universal-algebra/algebraic-theory-of-groups.lagda.md
@@ -146,7 +146,8 @@ abstract
   pr1 equiv-group-Algebra-Group = group-Algebra-Group
   pr1 (pr1 (pr2 equiv-group-Algebra-Group)) = Group-group-Algebra
   pr2 (pr1 (pr2 equiv-group-Algebra-Group)) G =
-    eq-pair-eq-fiber (eq-is-prop (is-prop-is-group (semigroup-Group G)))
+    eq-pair-eq-fiber
+      ( eq-is-prop (is-prop-is-group-Semigroup (semigroup-Group G)))
   pr1 (pr2 (pr2 equiv-group-Algebra-Group)) = Group-group-Algebra
   pr2 (pr2 (pr2 equiv-group-Algebra-Group)) A =
     eq-pair-Σ