mathlib3 documentation

category_theory.products.basic

Cartesian products of categories #

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We define the category instance on C × D when C and D are categories.

We define:

We further define evaluation : C ⥤ (C ⥤ D) ⥤ D and evaluation_uncurried : C × (C ⥤ D) ⥤ D, and products of functors and natural transformations, written F.prod G and α.prod β.

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@[simp]
theorem category_theory.prod_id_fst (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C × D) :
(𝟙 X).fst = 𝟙 X.fst
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@[simp]
theorem category_theory.prod_id_snd (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C × D) :
(𝟙 X).snd = 𝟙 X.snd
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@[simp]
theorem category_theory.prod_comp_fst (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (_x _x_1 _x_2 : C × D) (f : _x ⟶ _x_1) (g : _x_1 ⟶ _x_2) :
(f ≫ g).fst = f.fst ≫ g.fst
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@[protected, instance]
def category_theory.prod (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] :
category_theory.category (C × D)

prod C D gives the cartesian product of two categories.

See https://stacks.math.columbia.edu/tag/001K.

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@[simp]
theorem category_theory.prod_comp_snd (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (_x _x_1 _x_2 : C × D) (f : _x ⟶ _x_1) (g : _x_1 ⟶ _x_2) :
(f ≫ g).snd = f.snd ≫ g.snd
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@[simp]
theorem category_theory.prod_id (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C) (Y : D) :
𝟙 (X, Y) = (𝟙 X, 𝟙 Y)

Two rfl lemmas that cannot be generated by @[simps].

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@[simp]
theorem category_theory.prod_comp (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] {P Q R : C} {S T U : D} (f : (P, S) ⟶ (Q, T)) (g : (Q, T) ⟶ (R, U)) :
f ≫ g = (f.fst ≫ g.fst, f.snd ≫ g.snd)
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theorem category_theory.is_iso_prod_iff (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] {P Q : C} {S T : D} {f : (P, S) ⟶ (Q, T)} :
category_theory.is_iso f ↔ category_theory.is_iso f.fst ∧ category_theory.is_iso f.snd
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@[simp]
theorem category_theory.prod.eta_iso_hom {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (X : C × D) :
(category_theory.prod.eta_iso X).hom = (𝟙 (X.fst, X.snd).fst, 𝟙 (X.fst, X.snd).snd)
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@[simp]
theorem category_theory.prod.eta_iso_inv {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (X : C × D) :
(category_theory.prod.eta_iso X).inv = (𝟙 X.fst, 𝟙 X.snd)
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def category_theory.prod.eta_iso {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (X : C × D) :
(X.fst, X.snd) ≅ X

The isomorphism between (X.1, X.2) and X.

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def category_theory.iso.prod {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {P Q : C} {S T : D} (f : P ≅ Q) (g : S ≅ T) :
(P, S) ≅ (Q, T)

Construct an isomorphism in C × D out of two isomorphisms in C and D.

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@[simp]
theorem category_theory.iso.prod_hom {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {P Q : C} {S T : D} (f : P ≅ Q) (g : S ≅ T) :
(f.prod g).hom = (f.hom, g.hom)
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@[simp]
theorem category_theory.iso.prod_inv {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] {P Q : C} {S T : D} (f : P ≅ Q) (g : S ≅ T) :
(f.prod g).inv = (f.inv, g.inv)
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@[protected, instance]
def category_theory.uniform_prod (C : Type u₁) [category_theory.category C] (D : Type u₁) [category_theory.category D] :
category_theory.category (C × D)

prod.category.uniform C D is an additional instance specialised so both factors have the same universe levels. This helps typeclass resolution.

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@[simp]
theorem category_theory.prod.sectl_map (C : Type u₁) [category_theory.category C] {D : Type u₂} [category_theory.category D] (Z : D) (X Y : C) (f : X ⟶ Y) :
(category_theory.prod.sectl C Z).map f = (f, 𝟙 Z)
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def category_theory.prod.sectl (C : Type u₁) [category_theory.category C] {D : Type u₂} [category_theory.category D] (Z : D) :
C ⥤ C × D

sectl C Z is the functor C ⥤ C × D given by X ↦ (X, Z).

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@[simp]
theorem category_theory.prod.sectl_obj (C : Type u₁) [category_theory.category C] {D : Type u₂} [category_theory.category D] (Z : D) (X : C) :
(category_theory.prod.sectl C Z).obj X = (X, Z)
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@[simp]
theorem category_theory.prod.sectr_map {C : Type u₁} [category_theory.category C] (Z : C) (D : Type u₂) [category_theory.category D] (X Y : D) (f : X ⟶ Y) :
(category_theory.prod.sectr Z D).map f = (𝟙 Z, f)
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def category_theory.prod.sectr {C : Type u₁} [category_theory.category C] (Z : C) (D : Type u₂) [category_theory.category D] :
D ⥤ C × D

sectr Z D is the functor D ⥤ C × D given by Y ↦ (Z, Y) .

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theorem category_theory.prod.sectr_obj {C : Type u₁} [category_theory.category C] (Z : C) (D : Type u₂) [category_theory.category D] (X : D) :
(category_theory.prod.sectr Z D).obj X = (Z, X)
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def category_theory.prod.fst (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] :
C × D ⥤ C

fst is the functor (X, Y) ↦ X.

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@[simp]
theorem category_theory.prod.fst_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X Y : C × D) (f : X ⟶ Y) :
(category_theory.prod.fst C D).map f = f.fst
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@[simp]
theorem category_theory.prod.fst_obj (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C × D) :
(category_theory.prod.fst C D).obj X = X.fst
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@[simp]
theorem category_theory.prod.snd_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X Y : C × D) (f : X ⟶ Y) :
(category_theory.prod.snd C D).map f = f.snd
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@[simp]
theorem category_theory.prod.snd_obj (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C × D) :
(category_theory.prod.snd C D).obj X = X.snd
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def category_theory.prod.snd (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] :
C × D ⥤ D

snd is the functor (X, Y) ↦ Y.

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theorem category_theory.prod.swap_obj (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C × D) :
(category_theory.prod.swap C D).obj X = (X.snd, X.fst)
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@[simp]
theorem category_theory.prod.swap_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (_x _x_1 : C × D) (f : _x ⟶ _x_1) :
(category_theory.prod.swap C D).map f = (f.snd, f.fst)
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def category_theory.prod.swap (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] :
C × D ⥤ D × C

The functor swapping the factors of a cartesian product of categories, C × D ⥤ D × C.

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Instances for category_theory.prod.swap
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@[simp]
theorem category_theory.prod.symmetry_hom_app (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C × D) :
(category_theory.prod.symmetry C D).hom.app X = 𝟙 X
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def category_theory.prod.symmetry (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] :
category_theory.prod.swap C D ⋙ category_theory.prod.swap D C ≅ 𝟭 (C × D)

Swapping the factors of a cartesion product of categories twice is naturally isomorphic to the identity functor.

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theorem category_theory.prod.symmetry_inv_app (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C × D) :
(category_theory.prod.symmetry C D).inv.app X = 𝟙 X
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@[simp]
theorem category_theory.prod.braiding_unit_iso_inv_app (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C × D) :
(category_theory.prod.braiding C D).unit_iso.inv.app X = (𝟙 X.fst, 𝟙 X.snd) ≫ category_theory.eq_to_hom _ ≫ (𝟙 X.fst, 𝟙 X.snd) ≫ (𝟙 X.fst, 𝟙 X.snd) ≫ (𝟙 X.fst, 𝟙 X.snd) ≫ (𝟙 X.fst, 𝟙 X.snd) ≫ category_theory.eq_to_hom _
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@[simp]
theorem category_theory.prod.braiding_functor_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (_x _x_1 : C × D) (f : _x ⟶ _x_1) :
(category_theory.prod.braiding C D).functor.map f = (f.snd, f.fst)
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@[simp]
theorem category_theory.prod.braiding_inverse_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (_x _x_1 : D × C) (f : _x ⟶ _x_1) :
(category_theory.prod.braiding C D).inverse.map f = (f.snd, f.fst)
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@[simp]
theorem category_theory.prod.braiding_counit_iso_hom_app (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : D × C) :
(category_theory.prod.braiding C D).counit_iso.hom.app X = category_theory.eq_to_hom _
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def category_theory.prod.braiding (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] :
C × D ≌ D × C

The equivalence, given by swapping factors, between C × D and D × C.

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theorem category_theory.prod.braiding_inverse_obj (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : D × C) :
(category_theory.prod.braiding C D).inverse.obj X = (X.snd, X.fst)
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@[simp]
theorem category_theory.prod.braiding_functor_obj (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C × D) :
(category_theory.prod.braiding C D).functor.obj X = (X.snd, X.fst)
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@[simp]
theorem category_theory.prod.braiding_unit_iso_hom_app (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C × D) :
(category_theory.prod.braiding C D).unit_iso.hom.app X = category_theory.eq_to_hom _ ≫ category_theory.eq_to_hom _
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@[simp]
theorem category_theory.prod.braiding_counit_iso_inv_app (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : D × C) :
(category_theory.prod.braiding C D).counit_iso.inv.app X = category_theory.eq_to_hom _
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@[protected, instance]
def category_theory.prod.swap_is_equivalence (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] :
category_theory.is_equivalence (category_theory.prod.swap C D)
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theorem category_theory.evaluation_map_app (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X Y : C) (f : X ⟶ Y) (F : C ⥤ D) :
((category_theory.evaluation C D).map f).app F = F.map f
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def category_theory.evaluation (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] :
C ⥤ (C ⥤ D) ⥤ D

The "evaluation at X" functor, such that (evaluation.obj X).obj F = F.obj X, which is functorial in both X and F.

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theorem category_theory.evaluation_obj_obj (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C) (F : C ⥤ D) :
((category_theory.evaluation C D).obj X).obj F = F.obj X
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@[simp]
theorem category_theory.evaluation_obj_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C) (F G : C ⥤ D) (α : F ⟶ G) :
((category_theory.evaluation C D).obj X).map α = α.app X
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@[simp]
theorem category_theory.evaluation_uncurried_map (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (x y : C × (C ⥤ D)) (f : x ⟶ y) :
(category_theory.evaluation_uncurried C D).map f = x.snd.map f.fst ≫ f.snd.app y.fst
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def category_theory.evaluation_uncurried (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] :
C × (C ⥤ D) ⥤ D

The "evaluation of F at X" functor, as a functor C × (C ⥤ D) ⥤ D.

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theorem category_theory.evaluation_uncurried_obj (C : Type u₁) [category_theory.category C] (D : Type u₂) [category_theory.category D] (p : C × (C ⥤ D)) :
(category_theory.evaluation_uncurried C D).obj p = p.snd.obj p.fst
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@[simp]
theorem category_theory.functor.const_comp_evaluation_obj_inv_app {C : Type u₁} [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C) (X_1 : D) :
(category_theory.functor.const_comp_evaluation_obj D X).inv.app X_1 = 𝟙 X_1
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def category_theory.functor.const_comp_evaluation_obj {C : Type u₁} [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C) :
category_theory.functor.const C ⋙ (category_theory.evaluation C D).obj X ≅ 𝟭 D

The constant functor followed by the evalutation functor is just the identity.

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theorem category_theory.functor.const_comp_evaluation_obj_hom_app {C : Type u₁} [category_theory.category C] (D : Type u₂) [category_theory.category D] (X : C) (X_1 : D) :
(category_theory.functor.const_comp_evaluation_obj D X).hom.app X_1 = 𝟙 X_1
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@[simp]
theorem category_theory.functor.prod_map {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] {D : Type u₄} [category_theory.category D] (F : A ⥤ B) (G : C ⥤ D) (_x _x_1 : A × C) (f : _x ⟶ _x_1) :
(F.prod G).map f = (F.map f.fst, G.map f.snd)
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@[simp]
theorem category_theory.functor.prod_obj {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] {D : Type u₄} [category_theory.category D] (F : A ⥤ B) (G : C ⥤ D) (X : A × C) :
(F.prod G).obj X = (F.obj X.fst, G.obj X.snd)
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def category_theory.functor.prod {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] {D : Type u₄} [category_theory.category D] (F : A ⥤ B) (G : C ⥤ D) :
A × C ⥤ B × D

The cartesian product of two functors.

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theorem category_theory.functor.prod'_map {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A ⥤ B) (G : A ⥤ C) (x y : A) (f : x ⟶ y) :
(F.prod' G).map f = (F.map f, G.map f)
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@[simp]
theorem category_theory.functor.prod'_obj {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A ⥤ B) (G : A ⥤ C) (a : A) :
(F.prod' G).obj a = (F.obj a, G.obj a)
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def category_theory.functor.prod' {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A ⥤ B) (G : A ⥤ C) :
A ⥤ B × C

Similar to prod, but both functors start from the same category A

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theorem category_theory.functor.prod'_comp_fst_inv_app {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A ⥤ B) (G : A ⥤ C) (X : A) :
(F.prod'_comp_fst G).inv.app X = 𝟙 (F.obj X)
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def category_theory.functor.prod'_comp_fst {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A ⥤ B) (G : A ⥤ C) :
F.prod' G ⋙ category_theory.prod.fst B C ≅ F

The product F.prod' G followed by projection on the first component is isomorphic to F

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theorem category_theory.functor.prod'_comp_fst_hom_app {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A ⥤ B) (G : A ⥤ C) (X : A) :
(F.prod'_comp_fst G).hom.app X = 𝟙 (F.obj X)
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@[simp]
theorem category_theory.functor.prod'_comp_snd_hom_app {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A ⥤ B) (G : A ⥤ C) (X : A) :
(F.prod'_comp_snd G).hom.app X = 𝟙 (G.obj X)
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def category_theory.functor.prod'_comp_snd {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A ⥤ B) (G : A ⥤ C) :
F.prod' G ⋙ category_theory.prod.snd B C ≅ G

The product F.prod' G followed by projection on the second component is isomorphic to G

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theorem category_theory.functor.prod'_comp_snd_inv_app {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A ⥤ B) (G : A ⥤ C) (X : A) :
(F.prod'_comp_snd G).inv.app X = 𝟙 (G.obj X)
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def category_theory.functor.diag (C : Type u₃) [category_theory.category C] :
C ⥤ C × C

The diagonal functor.

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theorem category_theory.functor.diag_obj (C : Type u₃) [category_theory.category C] (X : C) :
(category_theory.functor.diag C).obj X = (X, X)
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@[simp]
theorem category_theory.functor.diag_map (C : Type u₃) [category_theory.category C] {X Y : C} (f : X ⟶ Y) :
(category_theory.functor.diag C).map f = (f, f)
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def category_theory.nat_trans.prod {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] {D : Type u₄} [category_theory.category D] {F G : A ⥤ B} {H I : C ⥤ D} (α : F ⟶ G) (β : H ⟶ I) :
F.prod H ⟶ G.prod I

The cartesian product of two natural transformations.

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@[simp]
theorem category_theory.nat_trans.prod_app {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] {D : Type u₄} [category_theory.category D] {F G : A ⥤ B} {H I : C ⥤ D} (α : F ⟶ G) (β : H ⟶ I) (X : A × C) :
(category_theory.nat_trans.prod α β).app X = (α.app X.fst, β.app X.snd)
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theorem category_theory.flip_comp_evaluation_hom_app {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A ⥤ B ⥤ C) (a : A) (X : B) :
(category_theory.flip_comp_evaluation F a).hom.app X = 𝟙 ((F.obj a).obj X)
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@[simp]
theorem category_theory.flip_comp_evaluation_inv_app {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A ⥤ B ⥤ C) (a : A) (X : B) :
(category_theory.flip_comp_evaluation F a).inv.app X = 𝟙 ((F.obj a).obj X)
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def category_theory.flip_comp_evaluation {A : Type u₁} [category_theory.category A] {B : Type u₂} [category_theory.category B] {C : Type u₃} [category_theory.category C] (F : A ⥤ B ⥤ C) (a : A) :
F.flip ⋙ (category_theory.evaluation A C).obj a ≅ F.obj a

F.flip composed with evaluation is the same as evaluating F.

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theorem category_theory.prod_functor_to_functor_prod_obj (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] (F : (A ⥤ B) × (A ⥤ C)) :
(category_theory.prod_functor_to_functor_prod A B C).obj F = F.fst.prod' F.snd
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@[simp]
theorem category_theory.prod_functor_to_functor_prod_map_app (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] (F G : (A ⥤ B) × (A ⥤ C)) (f : F ⟶ G) (X : A) :
((category_theory.prod_functor_to_functor_prod A B C).map f).app X = (f.fst.app X, f.snd.app X)
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def category_theory.prod_functor_to_functor_prod (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] :
(A ⥤ B) × (A ⥤ C) ⥤ A ⥤ B × C

The forward direction for functor_prod_functor_equiv

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@[simp]
theorem category_theory.functor_prod_to_prod_functor_obj (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] (F : A ⥤ B × C) :
(category_theory.functor_prod_to_prod_functor A B C).obj F = (F ⋙ category_theory.prod.fst B C, F ⋙ category_theory.prod.snd B C)
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def category_theory.functor_prod_to_prod_functor (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] :
(A ⥤ B × C) ⥤ (A ⥤ B) × (A ⥤ C)

The backward direction for functor_prod_functor_equiv

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@[simp]
theorem category_theory.functor_prod_to_prod_functor_map (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] (F G : A ⥤ B × C) (α : F ⟶ G) :
(category_theory.functor_prod_to_prod_functor A B C).map α = ({app := λ (X : A), (α.app X).fst, naturality' := _}, {app := λ (X : A), (α.app X).snd, naturality' := _})
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def category_theory.functor_prod_functor_equiv_unit_iso (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] :
𝟭 ((A ⥤ B) × (A ⥤ C)) ≅ category_theory.prod_functor_to_functor_prod A B C ⋙ category_theory.functor_prod_to_prod_functor A B C

The unit isomorphism for functor_prod_functor_equiv

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@[simp]
theorem category_theory.functor_prod_functor_equiv_unit_iso_hom_app (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] (X : (A ⥤ B) × (A ⥤ C)) :
(category_theory.functor_prod_functor_equiv_unit_iso A B C).hom.app X = (𝟙 X.fst, 𝟙 X.snd) ≫ ((X.fst.prod'_comp_fst X.snd).inv, (X.fst.prod'_comp_snd X.snd).inv)
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@[simp]
theorem category_theory.functor_prod_functor_equiv_unit_iso_inv_app (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] (X : (A ⥤ B) × (A ⥤ C)) :
(category_theory.functor_prod_functor_equiv_unit_iso A B C).inv.app X = ((X.fst.prod'_comp_fst X.snd).hom, (X.fst.prod'_comp_snd X.snd).hom) ≫ (𝟙 X.fst, 𝟙 X.snd)
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@[simp]
theorem category_theory.functor_prod_functor_equiv_counit_iso_hom_app_app (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] (X : A ⥤ B × C) (X_1 : A) :
((category_theory.functor_prod_functor_equiv_counit_iso A B C).hom.app X).app X_1 = (𝟙 (X.obj X_1).fst, 𝟙 (X.obj X_1).snd)
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@[simp]
theorem category_theory.functor_prod_functor_equiv_counit_iso_inv_app_app (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] (X : A ⥤ B × C) (X_1 : A) :
((category_theory.functor_prod_functor_equiv_counit_iso A B C).inv.app X).app X_1 = (𝟙 (X.obj X_1).fst, 𝟙 (X.obj X_1).snd)
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def category_theory.functor_prod_functor_equiv_counit_iso (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] :
category_theory.functor_prod_to_prod_functor A B C ⋙ category_theory.prod_functor_to_functor_prod A B C ≅ 𝟭 (A ⥤ B × C)

The counit isomorphism for functor_prod_functor_equiv

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@[simp]
theorem category_theory.functor_prod_functor_equiv_functor (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] :
(category_theory.functor_prod_functor_equiv A B C).functor = category_theory.prod_functor_to_functor_prod A B C
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@[simp]
theorem category_theory.functor_prod_functor_equiv_unit_iso_2 (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] :
(category_theory.functor_prod_functor_equiv A B C).unit_iso = category_theory.functor_prod_functor_equiv_unit_iso A B C
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def category_theory.functor_prod_functor_equiv (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] :
(A ⥤ B) × (A ⥤ C) ≌ A ⥤ B × C

The equivalence of categories between (A ⥤ B) × (A ⥤ C) and A ⥤ (B × C)

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theorem category_theory.functor_prod_functor_equiv_inverse (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] :
(category_theory.functor_prod_functor_equiv A B C).inverse = category_theory.functor_prod_to_prod_functor A B C
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@[simp]
theorem category_theory.functor_prod_functor_equiv_counit_iso_2 (A : Type u₁) [category_theory.category A] (B : Type u₂) [category_theory.category B] (C : Type u₃) [category_theory.category C] :
(category_theory.functor_prod_functor_equiv A B C).counit_iso = category_theory.functor_prod_functor_equiv_counit_iso A B C