diff --git a/Mathlib.lean b/Mathlib.lean
index 8ca5515ecafdc..ceb98def06822 100644
--- a/Mathlib.lean
+++ b/Mathlib.lean
@@ -257,6 +257,7 @@ import Mathlib.CategoryTheory.Functor.InvIsos
 import Mathlib.CategoryTheory.Groupoid
 import Mathlib.CategoryTheory.Groupoid.Basic
 import Mathlib.CategoryTheory.Iso
+import Mathlib.CategoryTheory.IsomorphismClasses
 import Mathlib.CategoryTheory.LiftingProperties.Basic
 import Mathlib.CategoryTheory.Monoidal.Category
 import Mathlib.CategoryTheory.NatIso
diff --git a/Mathlib/CategoryTheory/IsomorphismClasses.lean b/Mathlib/CategoryTheory/IsomorphismClasses.lean
new file mode 100644
index 0000000000000..de9d49621d3bf
--- /dev/null
+++ b/Mathlib/CategoryTheory/IsomorphismClasses.lean
@@ -0,0 +1,72 @@
+/-
+Copyright (c) 2019 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+
+! This file was ported from Lean 3 source module category_theory.isomorphism_classes
+! leanprover-community/mathlib commit 28aa996fc6fb4317f0083c4e6daf79878d81be33
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.CategoryTheory.Category.Cat
+import Mathlib.CategoryTheory.Groupoid
+import Mathlib.CategoryTheory.Types
+
+/-!
+# Objects of a category up to an isomorphism
+
+`IsIsomorphic X Y := Nonempty (X ≅ Y)` is an equivalence relation on the objects of a category.
+The quotient with respect to this relation defines a functor from our category to `Type`.
+-/
+
+
+universe v u
+
+namespace CategoryTheory
+
+section Category
+
+variable {C : Type u} [Category.{v} C]
+
+/-- An object `X` is isomorphic to an object `Y`, if `X ≅ Y` is not empty. -/
+def IsIsomorphic : C → C → Prop := fun X Y => Nonempty (X ≅ Y)
+#align category_theory.is_isomorphic CategoryTheory.IsIsomorphic
+
+variable (C)
+
+/-- `is_isomorphic` defines a setoid. -/
+def isIsomorphicSetoid : Setoid C where
+  r := IsIsomorphic
+  iseqv := ⟨fun X => ⟨Iso.refl X⟩, fun ⟨α⟩ => ⟨α.symm⟩, fun ⟨α⟩ ⟨β⟩ => ⟨α.trans β⟩⟩
+#align category_theory.is_isomorphic_setoid CategoryTheory.isIsomorphicSetoid
+
+end Category
+
+/-- The functor that sends each category to the quotient space of its objects up to an isomorphism.
+-/
+def isomorphismClasses : Cat.{v, u} ⥤ Type u where
+  obj C := Quotient (isIsomorphicSetoid C.α)
+  map {C} {D} F := Quot.map F.obj fun X Y ⟨f⟩ => ⟨F.mapIso f⟩
+  map_id {C} := by  -- Porting note: this used to be `tidy`
+    dsimp; apply funext; intro x
+    apply x.recOn  -- Porting note: `induction x` not working yet 
+    · intro _ _ p 
+      simp only [types_id_apply]
+    · intro _ 
+      rfl 
+  map_comp {C} {D} {E} f g := by -- Porting note(s): idem
+    dsimp; apply funext; intro x
+    apply x.recOn 
+    · intro _ _ _ 
+      simp only [types_id_apply]
+    · intro _
+      rfl 
+#align category_theory.isomorphism_classes CategoryTheory.isomorphismClasses
+
+theorem Groupoid.isIsomorphic_iff_nonempty_hom {C : Type u} [Groupoid.{v} C] {X Y : C} :
+    IsIsomorphic X Y ↔ Nonempty (X ⟶ Y) :=
+  (Groupoid.isoEquivHom X Y).nonempty_congr
+#align category_theory.groupoid.is_isomorphic_iff_nonempty_hom CategoryTheory.Groupoid.isIsomorphic_iff_nonempty_hom
+
+end CategoryTheory
+