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category-of-maps-categories.lagda.md
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category-of-maps-categories.lagda.md
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# The category of maps and natural transformations between two categories
```agda
module category-theory.category-of-maps-categories where
```
<details><summary>Imports</summary>
```agda
open import category-theory.categories
open import category-theory.commuting-squares-of-morphisms-in-precategories
open import category-theory.isomorphisms-in-categories
open import category-theory.isomorphisms-in-precategories
open import category-theory.maps-categories
open import category-theory.maps-precategories
open import category-theory.natural-isomorphisms-maps-categories
open import category-theory.natural-isomorphisms-maps-precategories
open import category-theory.natural-transformations-maps-precategories
open import category-theory.precategories
open import category-theory.precategory-of-maps-precategories
open import foundation.action-on-identifications-binary-functions
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.function-types
open import foundation.functoriality-dependent-function-types
open import foundation.functoriality-dependent-pair-types
open import foundation.identity-types
open import foundation.type-arithmetic-dependent-pair-types
open import foundation.type-theoretic-principle-of-choice
open import foundation.univalence
open import foundation.universe-levels
```
</details>
## Idea
[Maps](category-theory.maps-categories.md) between
[categories](category-theory.categories.md) and
[natural transformations](category-theory.natural-transformations-maps-categories.md)
between them form another category whose identity map and composition structure
are inherited pointwise from the codomain category. This is called the
**category of maps between categories**.
## Lemmas
### Extensionality of maps of precategories when the codomain is a category
```agda
module _
{l1 l2 l3 l4 : Level}
(C : Precategory l1 l2)
(D : Precategory l3 l4)
(is-category-D : is-category-Precategory D)
where
equiv-natural-isomorphism-htpy-map-is-category-Precategory :
(F G : map-Precategory C D) →
( htpy-map-Precategory C D F G) ≃
( natural-isomorphism-map-Precategory C D F G)
equiv-natural-isomorphism-htpy-map-is-category-Precategory F G =
( equiv-right-swap-Σ) ∘e
( equiv-Σ
( is-natural-transformation-map-Precategory C D F G ∘ pr1)
( ( distributive-Π-Σ) ∘e
( equiv-Π-equiv-family
( λ x →
extensionality-obj-Category
( D , is-category-D)
( obj-map-Precategory C D F x)
( obj-map-Precategory C D G x))))
( λ K →
equiv-implicit-Π-equiv-family
( λ x →
equiv-implicit-Π-equiv-family
( λ y →
equiv-Π-equiv-family
( λ a →
( equiv-eq
( ap-binary
( λ p q →
coherence-square-hom-Precategory D
( hom-map-Precategory C D F a)
( p)
( q)
( hom-map-Precategory C D G a))
( compute-hom-iso-eq-Precategory D (K x))
( compute-hom-iso-eq-Precategory D (K y)))))))))
extensionality-map-is-category-Precategory :
(F G : map-Precategory C D) →
( F ＝ G) ≃
( natural-isomorphism-map-Precategory C D F G)
extensionality-map-is-category-Precategory F G =
( equiv-natural-isomorphism-htpy-map-is-category-Precategory F G) ∘e
( equiv-htpy-eq-map-Precategory C D F G)
```
### When the codomain is a category the map precategory is a category
```agda
module _
{l1 l2 l3 l4 : Level}
(C : Precategory l1 l2)
(D : Precategory l3 l4)
(is-category-D : is-category-Precategory D)
where
abstract
is-category-map-precategory-is-category-Precategory :
is-category-Precategory (map-precategory-Precategory C D)
is-category-map-precategory-is-category-Precategory F G =
is-equiv-htpy-equiv
( ( equiv-iso-map-natural-isomorphism-map-Precategory C D F G) ∘e
( extensionality-map-is-category-Precategory C D is-category-D F G))
( λ where
refl →
compute-iso-map-natural-isomorphism-map-eq-Precategory C D F G refl)
```
## Definitions
### The category of maps and natural transformations between categories
```agda
module _
{l1 l2 l3 l4 : Level}
(C : Category l1 l2) (D : Category l3 l4)
where
map-category-Category :
Category (l1 ⊔ l2 ⊔ l3 ⊔ l4) (l1 ⊔ l2 ⊔ l4)
pr1 map-category-Category =
map-precategory-Precategory
( precategory-Category C)
( precategory-Category D)
pr2 map-category-Category =
is-category-map-precategory-is-category-Precategory
( precategory-Category C)
( precategory-Category D)
( is-category-Category D)
extensionality-map-Category :
(F G : map-Category C D) →
(F ＝ G) ≃ natural-isomorphism-map-Category C D F G
extensionality-map-Category F G =
( equiv-natural-isomorphism-map-iso-map-Precategory
( precategory-Category C)
( precategory-Category D) F G) ∘e
( extensionality-obj-Category map-category-Category F G)
eq-natural-isomorphism-map-Category :
(F G : map-Category C D) →
natural-isomorphism-map-Category C D F G → F ＝ G
eq-natural-isomorphism-map-Category F G =
map-inv-equiv (extensionality-map-Category F G)
```