Properties of morphisms

We provide the basic framework for talking about properties of morphisms. The following meta-property is defined

• RespectsLeft P Q: P respects the property Q on the left if P f → P (i ≫ f) where i satisfies Q.
• RespectsRight P Q: P respects the property Q on the right if P f → P (f ≫ i) where i satisfies Q.
• Respects: P respects Q if P respects Q both on the left and on the right.

def CategoryTheory.MorphismProperty (C : Type u) [Category.{v, u} C] : Type (max u v)

A MorphismProperty C is a class of morphisms between objects in C.

Equations

• CategoryTheory.MorphismProperty C = (⦃X Y : C⦄ → (X ⟶ Y) → Prop)

instance CategoryTheory.instCompleteBooleanAlgebraMorphismProperty (C : Type u) [Category.{v, u} C] : CompleteBooleanAlgebra (MorphismProperty C)

Equations

• CategoryTheory.instCompleteBooleanAlgebraMorphismProperty C = CompleteBooleanAlgebra.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

theorem CategoryTheory.MorphismProperty.le_def (C : Type u) [Category.{v, u} C] {P Q : MorphismProperty C} : P ≤ Q ↔ ∀ {X Y : C} (f : X ⟶ Y), P f → Q f

instance CategoryTheory.instInhabitedMorphismProperty (C : Type u) [Category.{v, u} C] : Inhabited (MorphismProperty C)

Equations

• CategoryTheory.instInhabitedMorphismProperty C = { default := ⊤ }

theorem CategoryTheory.MorphismProperty.top_eq (C : Type u) [Category.{v, u} C] : ⊤ = fun (x x_1 : C) (x : x ⟶ x_1) => True

theorem CategoryTheory.MorphismProperty.ext {C : Type u} [Category.{v, u} C] (W W' : MorphismProperty C) (h : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f ↔ W' f) : W = W'

@[simp]

theorem CategoryTheory.MorphismProperty.top_apply {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) : ⊤ f

def CategoryTheory.MorphismProperty.op {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) : MorphismProperty Cᵒᵖ

The morphism property in Cᵒᵖ associated to a morphism property in C

Equations

• P.op f = P f.unop

def CategoryTheory.MorphismProperty.unop {C : Type u} [Category.{v, u} C] (P : MorphismProperty Cᵒᵖ) : MorphismProperty C

The morphism property in C associated to a morphism property in Cᵒᵖ

Equations

• P.unop f = P f.op

theorem CategoryTheory.MorphismProperty.unop_op {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) : P.op.unop = P

theorem CategoryTheory.MorphismProperty.op_unop {C : Type u} [Category.{v, u} C] (P : MorphismProperty Cᵒᵖ) : P.unop.op = P

def CategoryTheory.MorphismProperty.inverseImage {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty D) (F : Functor C D) : MorphismProperty C

The inverse image of a MorphismProperty D by a functor C ⥤ D

Equations

• P.inverseImage F f = P (F.map f)

@[simp]

theorem CategoryTheory.MorphismProperty.inverseImage_iff {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty D) (F : Functor C D) {X Y : C} (f : X ⟶ Y) : P.inverseImage F f ↔ P (F.map f)

def CategoryTheory.MorphismProperty.map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty C) (F : Functor C D) : MorphismProperty D

The image (up to isomorphisms) of a MorphismProperty C by a functor C ⥤ D

Equations

• P.map F f = ∃ (X' : C) (Y' : C) (f' : X' ⟶ Y') (_ : P f'), Nonempty (CategoryTheory.Arrow.mk (F.map f') ≅ CategoryTheory.Arrow.mk f)

theorem CategoryTheory.MorphismProperty.map_mem_map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty C) (F : Functor C D) {X Y : C} (f : X ⟶ Y) (hf : P f) : P.map F (F.map f)

theorem CategoryTheory.MorphismProperty.monotone_map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (F : Functor C D) : Monotone fun (x : MorphismProperty C) => x.map F

theorem CategoryTheory.MorphismProperty.of_eq {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) {X Y : C} {f : X ⟶ Y} (hf : P f) {X' Y' : C} {f' : X' ⟶ Y'} (hX : X = X') (hY : Y = Y') (h : f' = CategoryStruct.comp (eqToHom ⋯) (CategoryStruct.comp f (eqToHom hY))) : P f'

class CategoryTheory.MorphismProperty.RespectsRight {C : Type u} [Category.{v, u} C] (P Q : MorphismProperty C) : Prop

A morphism property P satisfies P.RespectsRight Q if it is stable under post-composition with morphisms satisfying Q, i.e. whenever P holds for f and Q holds for i then P holds for f ≫ i.

• postcomp {X Y Z : C} (i : Y ⟶ Z) (hi : Q i) (f : X ⟶ Y) (hf : P f) : P (CategoryStruct.comp f i)

class CategoryTheory.MorphismProperty.RespectsLeft {C : Type u} [Category.{v, u} C] (P Q : MorphismProperty C) : Prop

A morphism property P satisfies P.RespectsLeft Q if it is stable under pre-composition with morphisms satisfying Q, i.e. whenever P holds for f and Q holds for i then P holds for i ≫ f.

• precomp {X Y Z : C} (i : X ⟶ Y) (hi : Q i) (f : Y ⟶ Z) (hf : P f) : P (CategoryStruct.comp i f)

class CategoryTheory.MorphismProperty.Respects {C : Type u} [Category.{v, u} C] (P Q : MorphismProperty C) extends P.RespectsLeft Q, P.RespectsRight Q : Prop

A morphism property P satisfies P.Respects Q if it is stable under composition on the left and right by morphisms satisfying Q.

• precomp {X Y Z : C} (i : X ⟶ Y) (hi : Q i) (f : Y ⟶ Z) (hf : P f) : P (CategoryStruct.comp i f)
• postcomp {X Y Z : C} (i : Y ⟶ Z) (hi : Q i) (f : X ⟶ Y) (hf : P f) : P (CategoryStruct.comp f i)

instance CategoryTheory.MorphismProperty.instRespectsOfRespectsLeftOfRespectsRight {C : Type u} [Category.{v, u} C] (P Q : MorphismProperty C) [P.RespectsLeft Q] [P.RespectsRight Q] : P.Respects Q

instance CategoryTheory.MorphismProperty.instRespectsRightOppositeOpOfRespectsLeft {C : Type u} [Category.{v, u} C] (P Q : MorphismProperty C) [P.RespectsLeft Q] : P.op.RespectsRight Q.op

instance CategoryTheory.MorphismProperty.instRespectsLeftOppositeOpOfRespectsRight {C : Type u} [Category.{v, u} C] (P Q : MorphismProperty C) [P.RespectsRight Q] : P.op.RespectsLeft Q.op

instance CategoryTheory.MorphismProperty.RespectsLeft.inf {C : Type u} [Category.{v, u} C] (P₁ P₂ Q : MorphismProperty C) [P₁.RespectsLeft Q] [P₂.RespectsLeft Q] : (P₁ ⊓ P₂).RespectsLeft Q

instance CategoryTheory.MorphismProperty.RespectsRight.inf {C : Type u} [Category.{v, u} C] (P₁ P₂ Q : MorphismProperty C) [P₁.RespectsRight Q] [P₂.RespectsRight Q] : (P₁ ⊓ P₂).RespectsRight Q

def CategoryTheory.MorphismProperty.isomorphisms (C : Type u) [Category.{v, u} C] : MorphismProperty C

The MorphismProperty C satisfied by isomorphisms in C.

Equations

• CategoryTheory.MorphismProperty.isomorphisms C f = CategoryTheory.IsIso f

def CategoryTheory.MorphismProperty.monomorphisms (C : Type u) [Category.{v, u} C] : MorphismProperty C

The MorphismProperty C satisfied by monomorphisms in C.

Equations

• CategoryTheory.MorphismProperty.monomorphisms C f = CategoryTheory.Mono f

def CategoryTheory.MorphismProperty.epimorphisms (C : Type u) [Category.{v, u} C] : MorphismProperty C

The MorphismProperty C satisfied by epimorphisms in C.

Equations

• CategoryTheory.MorphismProperty.epimorphisms C f = CategoryTheory.Epi f

@[reducible, inline]

abbrev CategoryTheory.MorphismProperty.RespectsIso {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) : Prop

P respects isomorphisms, if it respects the morphism property isomorphisms C, i.e. it is stable under pre- and postcomposition with isomorphisms.

Equations

• P.RespectsIso = P.Respects (CategoryTheory.MorphismProperty.isomorphisms C)

theorem CategoryTheory.MorphismProperty.RespectsIso.mk {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) (hprecomp : ∀ {X Y Z : C} (e : X ≅ Y) (f : Y ⟶ Z), P f → P (CategoryStruct.comp e.hom f)) (hpostcomp : ∀ {X Y Z : C} (e : Y ≅ Z) (f : X ⟶ Y), P f → P (CategoryStruct.comp f e.hom)) : P.RespectsIso

theorem CategoryTheory.MorphismProperty.RespectsIso.precomp {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [P.RespectsIso] {X Y Z : C} (e : X ⟶ Y) [IsIso e] (f : Y ⟶ Z) (hf : P f) : P (CategoryStruct.comp e f)

instance CategoryTheory.MorphismProperty.instRespectsIsoTop {C : Type u} [Category.{v, u} C] : ⊤.RespectsIso

theorem CategoryTheory.MorphismProperty.RespectsIso.postcomp {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [P.RespectsIso] {X Y Z : C} (e : Y ⟶ Z) [IsIso e] (f : X ⟶ Y) (hf : P f) : P (CategoryStruct.comp f e)

instance CategoryTheory.MorphismProperty.RespectsIso.op {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [P.RespectsIso] : P.op.RespectsIso

instance CategoryTheory.MorphismProperty.RespectsIso.unop {C : Type u} [Category.{v, u} C] (P : MorphismProperty Cᵒᵖ) [P.RespectsIso] : P.unop.RespectsIso

def CategoryTheory.MorphismProperty.isoClosure {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) : MorphismProperty C

The closure by isomorphisms of a MorphismProperty

Equations

• P.isoClosure f = ∃ (Y₁ : C) (Y₂ : C) (f' : Y₁ ⟶ Y₂) (_ : P f'), Nonempty (CategoryTheory.Arrow.mk f' ≅ CategoryTheory.Arrow.mk f)

theorem CategoryTheory.MorphismProperty.le_isoClosure {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) : P ≤ P.isoClosure

instance CategoryTheory.MorphismProperty.isoClosure_respectsIso {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) : P.isoClosure.RespectsIso

theorem CategoryTheory.MorphismProperty.monotone_isoClosure {C : Type u} [Category.{v, u} C] : Monotone isoClosure

theorem CategoryTheory.MorphismProperty.cancel_left_of_respectsIso {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [hP : P.RespectsIso] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] : P (CategoryStruct.comp f g) ↔ P g

theorem CategoryTheory.MorphismProperty.cancel_right_of_respectsIso {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [hP : P.RespectsIso] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso g] : P (CategoryStruct.comp f g) ↔ P f

theorem CategoryTheory.MorphismProperty.comma_iso_iff {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [P.RespectsIso] {A : Type u_2} {B : Type u_3} [Category.{u_4, u_2} A] [Category.{u_5, u_3} B] {L : Functor A C} {R : Functor B C} {f g : Comma L R} (e : f ≅ g) : P f.hom ↔ P g.hom

theorem CategoryTheory.MorphismProperty.arrow_iso_iff {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [P.RespectsIso] {f g : Arrow C} (e : f ≅ g) : P f.hom ↔ P g.hom

theorem CategoryTheory.MorphismProperty.arrow_mk_iso_iff {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [P.RespectsIso] {W X Y Z : C} {f : W ⟶ X} {g : Y ⟶ Z} (e : Arrow.mk f ≅ Arrow.mk g) : P f ↔ P g

theorem CategoryTheory.MorphismProperty.RespectsIso.of_respects_arrow_iso {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) (hP : ∀ (f g : Arrow C), (f ≅ g) → P f.hom → P g.hom) : P.RespectsIso

theorem CategoryTheory.MorphismProperty.isoClosure_eq_iff {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) : P.isoClosure = P ↔ P.RespectsIso

theorem CategoryTheory.MorphismProperty.isoClosure_eq_self {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [P.RespectsIso] : P.isoClosure = P

@[simp]

theorem CategoryTheory.MorphismProperty.isoClosure_isoClosure {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) : P.isoClosure.isoClosure = P.isoClosure

theorem CategoryTheory.MorphismProperty.isoClosure_le_iff {C : Type u} [Category.{v, u} C] (P Q : MorphismProperty C) [Q.RespectsIso] : P.isoClosure ≤ Q ↔ P ≤ Q

instance CategoryTheory.MorphismProperty.map_respectsIso {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty C) (F : Functor C D) : (P.map F).RespectsIso

theorem CategoryTheory.MorphismProperty.map_le_iff {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty C) {F : Functor C D} (Q : MorphismProperty D) [Q.RespectsIso] : P.map F ≤ Q ↔ P ≤ Q.inverseImage F

@[simp]

theorem CategoryTheory.MorphismProperty.map_isoClosure {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty C) (F : Functor C D) : P.isoClosure.map F = P.map F

theorem CategoryTheory.MorphismProperty.map_id_eq_isoClosure {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) : P.map (Functor.id C) = P.isoClosure

theorem CategoryTheory.MorphismProperty.map_id {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [P.RespectsIso] : P.map (Functor.id C) = P

@[simp]

theorem CategoryTheory.MorphismProperty.map_map {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_3, u_1} D] (P : MorphismProperty C) (F : Functor C D) {E : Type u_2} [Category.{u_4, u_2} E] (G : Functor D E) : (P.map F).map G = P.map (F.comp G)

instance CategoryTheory.MorphismProperty.RespectsIso.inverseImage {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty D) [P.RespectsIso] (F : Functor C D) : (P.inverseImage F).RespectsIso

theorem CategoryTheory.MorphismProperty.map_eq_of_iso {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty C) {F G : Functor C D} (e : F ≅ G) : P.map F = P.map G

theorem CategoryTheory.MorphismProperty.map_inverseImage_le {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty D) (F : Functor C D) : (P.inverseImage F).map F ≤ P.isoClosure

theorem CategoryTheory.MorphismProperty.inverseImage_equivalence_inverse_eq_map_functor {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty D) [P.RespectsIso] (E : C ≌ D) : P.inverseImage E.functor = P.map E.inverse

theorem CategoryTheory.MorphismProperty.inverseImage_equivalence_functor_eq_map_inverse {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (Q : MorphismProperty C) [Q.RespectsIso] (E : C ≌ D) : Q.inverseImage E.inverse = Q.map E.functor

theorem CategoryTheory.MorphismProperty.map_inverseImage_eq_of_isEquivalence {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty D) [P.RespectsIso] (F : Functor C D) [F.IsEquivalence] : (P.inverseImage F).map F = P

theorem CategoryTheory.MorphismProperty.inverseImage_map_eq_of_isEquivalence {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] (P : MorphismProperty C) [P.RespectsIso] (F : Functor C D) [F.IsEquivalence] : (P.map F).inverseImage F = P

@[simp]

theorem CategoryTheory.MorphismProperty.isomorphisms.iff {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) : isomorphisms C f ↔ IsIso f

@[simp]

theorem CategoryTheory.MorphismProperty.monomorphisms.iff {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) : monomorphisms C f ↔ Mono f

@[simp]

theorem CategoryTheory.MorphismProperty.epimorphisms.iff {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) : epimorphisms C f ↔ Epi f

theorem CategoryTheory.MorphismProperty.isomorphisms.infer_property {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [hf : IsIso f] : isomorphisms C f

theorem CategoryTheory.MorphismProperty.monomorphisms.infer_property {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [hf : Mono f] : monomorphisms C f

theorem CategoryTheory.MorphismProperty.epimorphisms.infer_property {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [hf : Epi f] : epimorphisms C f

instance CategoryTheory.MorphismProperty.RespectsIso.monomorphisms (C : Type u) [Category.{v, u} C] : (MorphismProperty.monomorphisms C).RespectsIso

instance CategoryTheory.MorphismProperty.RespectsIso.epimorphisms (C : Type u) [Category.{v, u} C] : (MorphismProperty.epimorphisms C).RespectsIso

instance CategoryTheory.MorphismProperty.RespectsIso.isomorphisms (C : Type u) [Category.{v, u} C] : (MorphismProperty.isomorphisms C).RespectsIso

@[deprecated CategoryTheory.MorphismProperty.cancel_left_of_respectsIso (since := "2024-07-02")]

theorem CategoryTheory.MorphismProperty.RespectsIso.cancel_left_isIso {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [hP : P.RespectsIso] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] : P (CategoryStruct.comp f g) ↔ P g

Alias of CategoryTheory.MorphismProperty.cancel_left_of_respectsIso.

@[deprecated CategoryTheory.MorphismProperty.cancel_right_of_respectsIso (since := "2024-07-02")]

theorem CategoryTheory.MorphismProperty.RespectsIso.cancel_right_isIso {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [hP : P.RespectsIso] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso g] : P (CategoryStruct.comp f g) ↔ P f

Alias of CategoryTheory.MorphismProperty.cancel_right_of_respectsIso.

@[deprecated CategoryTheory.MorphismProperty.arrow_iso_iff (since := "2024-07-02")]

theorem CategoryTheory.MorphismProperty.RespectsIso.arrow_iso_iff {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [P.RespectsIso] {f g : Arrow C} (e : f ≅ g) : P f.hom ↔ P g.hom

Alias of CategoryTheory.MorphismProperty.arrow_iso_iff.

@[deprecated CategoryTheory.MorphismProperty.arrow_mk_iso_iff (since := "2024-07-02")]

theorem CategoryTheory.MorphismProperty.RespectsIso.arrow_mk_iso_iff {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [P.RespectsIso] {W X Y Z : C} {f : W ⟶ X} {g : Y ⟶ Z} (e : Arrow.mk f ≅ Arrow.mk g) : P f ↔ P g

Alias of CategoryTheory.MorphismProperty.arrow_mk_iso_iff.

@[deprecated CategoryTheory.MorphismProperty.isoClosure_eq_self (since := "2024-07-02")]

theorem CategoryTheory.MorphismProperty.RespectsIso.isoClosure_eq {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [P.RespectsIso] : P.isoClosure = P

Alias of CategoryTheory.MorphismProperty.isoClosure_eq_self.

def CategoryTheory.MorphismProperty.prod {C₁ : Type u_2} {C₂ : Type u_3} [Category.{u_4, u_2} C₁] [Category.{u_5, u_3} C₂] (W₁ : MorphismProperty C₁) (W₂ : MorphismProperty C₂) : MorphismProperty (C₁ × C₂)

If W₁ and W₂ are morphism properties on two categories C₁ and C₂, this is the induced morphism property on C₁ × C₂.

Equations

• W₁.prod W₂ f = (W₁ f.1 ∧ W₂ f.2)

def CategoryTheory.MorphismProperty.pi {J : Type w} {C : J → Type u} [(j : J) → Category.{v, u} (C j)] (W : (j : J) → MorphismProperty (C j)) : MorphismProperty ((j : J) → C j)

If W j are morphism properties on categories C j for all j, this is the induced morphism property on the category ∀ j, C j.

Equations

• CategoryTheory.MorphismProperty.pi W f = ∀ (j : J), W j (f j)

def CategoryTheory.MorphismProperty.functorCategory {C : Type u} [Category.{v, u} C] (W : MorphismProperty C) (J : Type u_2) [Category.{u_3, u_2} J] : MorphismProperty (Functor J C)

The morphism property on J ⥤ C which is defined objectwise from W : MorphismProperty C.

Equations

• W.functorCategory J f = ∀ (j : J), W (f.app j)

def CategoryTheory.MorphismProperty.arrow {C : Type u} [Category.{v, u} C] (W : MorphismProperty C) : MorphismProperty (Arrow C)

Given W : MorphismProperty C, this is the morphism property on Arrow C of morphisms whose left and right parts are in W.

Equations

• W.arrow f = (W f.left ∧ W f.right)

theorem CategoryTheory.NatTrans.isIso_app_iff_of_iso {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{u_2, u_1} D] {F G : Functor C D} (α : F ⟶ G) {X Y : C} (e : X ≅ Y) : IsIso (α.app X) ↔ IsIso (α.app Y)