Curry and uncurry, as functors.

We define curry : ((C × D) ⥤ E) ⥤ (C ⥤ (D ⥤ E)) and uncurry : (C ⥤ (D ⥤ E)) ⥤ ((C × D) ⥤ E), and verify that they provide an equivalence of categories currying : (C ⥤ (D ⥤ E)) ≌ ((C × D) ⥤ E).

def CategoryTheory.uncurry {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] : CategoryTheory.Functor (CategoryTheory.Functor C (CategoryTheory.Functor D E)) (CategoryTheory.Functor (C × D) E)

The uncurrying functor, taking a functor C ⥤ (D ⥤ E) and producing a functor (C × D) ⥤ E.

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theorem CategoryTheory.uncurry_obj_map {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) {X : C × D} {Y : C × D} (f : X ⟶ Y) : (CategoryTheory.uncurry.obj F).map f = CategoryTheory.CategoryStruct.comp ((F.map f.1).app X.2) ((F.obj Y.1).map f.2)

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theorem CategoryTheory.uncurry_obj_obj {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : C × D) : (CategoryTheory.uncurry.obj F).obj X = (F.obj X.1).obj X.2

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theorem CategoryTheory.uncurry_map_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] : ∀ {X Y : CategoryTheory.Functor C (CategoryTheory.Functor D E)} (T : X ⟶ Y) (X_1 : C × D), (CategoryTheory.uncurry.map T).app X_1 = (T.app X_1.1).app X_1.2

def CategoryTheory.curryObj {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) : CategoryTheory.Functor C (CategoryTheory.Functor D E)

The object level part of the currying functor. (See curry for the functorial version.)

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def CategoryTheory.curry {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] : CategoryTheory.Functor (CategoryTheory.Functor (C × D) E) (CategoryTheory.Functor C (CategoryTheory.Functor D E))

The currying functor, taking a functor (C × D) ⥤ E and producing a functor C ⥤ (D ⥤ E).

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theorem CategoryTheory.curry_obj_obj_map {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) (X : C) : ∀ {X_1 Y : D} (g : X_1 ⟶ Y), ((CategoryTheory.curry.obj F).obj X).map g = F.map (CategoryTheory.CategoryStruct.id X, g)

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theorem CategoryTheory.curry_obj_obj_obj {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) (X : C) (Y : D) : ((CategoryTheory.curry.obj F).obj X).obj Y = F.obj (X, Y)

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theorem CategoryTheory.curry_obj_map_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) : ∀ {X Y : C} (f : X ⟶ Y) (Y_1 : D), ((CategoryTheory.curry.obj F).map f).app Y_1 = F.map (f, CategoryTheory.CategoryStruct.id Y_1)

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theorem CategoryTheory.curry_map_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] : ∀ {X Y : CategoryTheory.Functor (C × D) E} (T : X ⟶ Y) (X_1 : C) (Y_1 : D), ((CategoryTheory.curry.map T).app X_1).app Y_1 = T.app (X_1, Y_1)

def CategoryTheory.currying {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] : CategoryTheory.Functor C (CategoryTheory.Functor D E) ≌ CategoryTheory.Functor (C × D) E

The equivalence of functor categories given by currying/uncurrying.

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theorem CategoryTheory.currying_inverse_obj_obj_obj {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) (X : C) (Y : D) : ((CategoryTheory.currying.inverse.obj F).obj X).obj Y = F.obj (X, Y)

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theorem CategoryTheory.currying_inverse_map_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] : ∀ {X Y : CategoryTheory.Functor (C × D) E} (T : X ⟶ Y) (X_1 : C) (Y_1 : D), ((CategoryTheory.currying.inverse.map T).app X_1).app Y_1 = T.app (X_1, Y_1)

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theorem CategoryTheory.currying_unitIso_inv_app_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : C) (X : D) : ((CategoryTheory.currying.unitIso.inv.app X✝¹).app X✝).app X = CategoryTheory.CategoryStruct.id ((X✝¹.obj X✝).obj X)

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theorem CategoryTheory.currying_unitIso_hom_app_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : C) (X : D) : ((CategoryTheory.currying.unitIso.hom.app X✝¹).app X✝).app X = CategoryTheory.CategoryStruct.id ((X✝¹.obj X✝).obj X)

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theorem CategoryTheory.currying_functor_obj_obj {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : C × D) : (CategoryTheory.currying.functor.obj F).obj X = (F.obj X.1).obj X.2

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theorem CategoryTheory.currying_functor_map_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] : ∀ {X Y : CategoryTheory.Functor C (CategoryTheory.Functor D E)} (T : X ⟶ Y) (X_1 : C × D), (CategoryTheory.currying.functor.map T).app X_1 = (T.app X_1.1).app X_1.2

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theorem CategoryTheory.currying_inverse_obj_obj_map {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) (X : C) : ∀ {X_1 Y : D} (g : X_1 ⟶ Y), ((CategoryTheory.currying.inverse.obj F).obj X).map g = F.map (CategoryTheory.CategoryStruct.id X, g)

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theorem CategoryTheory.currying_inverse_obj_map_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) : ∀ {X Y : C} (f : X ⟶ Y) (Y_1 : D), ((CategoryTheory.currying.inverse.obj F).map f).app Y_1 = F.map (f, CategoryTheory.CategoryStruct.id Y_1)

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theorem CategoryTheory.currying_functor_obj_map {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) {X : C × D} {Y : C × D} (f : X ⟶ Y) : (CategoryTheory.currying.functor.obj F).map f = CategoryTheory.CategoryStruct.comp ((F.map f.1).app X.2) ((F.obj Y.1).map f.2)

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theorem CategoryTheory.currying_counitIso_inv_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor (C × D) E) (X : C × D) : (CategoryTheory.currying.counitIso.inv.app X✝).app X = CategoryTheory.CategoryStruct.id (X✝.obj (X.1, X.2))

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theorem CategoryTheory.currying_counitIso_hom_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor (C × D) E) (X : C × D) : (CategoryTheory.currying.counitIso.hom.app X✝).app X = CategoryTheory.CategoryStruct.id (X✝.obj (X.1, X.2))

def CategoryTheory.flipIsoCurrySwapUncurry {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) : F.flip ≅ CategoryTheory.curry.obj ((CategoryTheory.Prod.swap D C).comp (CategoryTheory.uncurry.obj F))

F.flip is isomorphic to uncurrying F, swapping the variables, and currying.

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theorem CategoryTheory.flipIsoCurrySwapUncurry_inv_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D) (X : C) : ((CategoryTheory.flipIsoCurrySwapUncurry F).inv.app X✝).app X = CategoryTheory.CategoryStruct.id ((F.obj X).obj X✝)

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theorem CategoryTheory.flipIsoCurrySwapUncurry_hom_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D) (X : C) : ((CategoryTheory.flipIsoCurrySwapUncurry F).hom.app X✝).app X = CategoryTheory.CategoryStruct.id ((F.obj X).obj X✝)

def CategoryTheory.uncurryObjFlip {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) : CategoryTheory.uncurry.obj F.flip ≅ (CategoryTheory.Prod.swap D C).comp (CategoryTheory.uncurry.obj F)

The uncurrying of F.flip is isomorphic to swapping the factors followed by the uncurrying of F.

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• CategoryTheory.uncurryObjFlip F = CategoryTheory.NatIso.ofComponents (fun (x : D × C) => CategoryTheory.Iso.refl ((CategoryTheory.uncurry.obj F.flip).obj x)) ⋯

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theorem CategoryTheory.uncurryObjFlip_inv_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D × C) : (CategoryTheory.uncurryObjFlip F).inv.app X = CategoryTheory.CategoryStruct.id ((F.obj X.2).obj X.1)

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theorem CategoryTheory.uncurryObjFlip_hom_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D × C) : (CategoryTheory.uncurryObjFlip F).hom.app X = CategoryTheory.CategoryStruct.id ((F.obj X.2).obj X.1)

def CategoryTheory.whiskeringRight₂ (B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] : CategoryTheory.Functor (CategoryTheory.Functor C (CategoryTheory.Functor D E)) (CategoryTheory.Functor (CategoryTheory.Functor B C) (CategoryTheory.Functor (CategoryTheory.Functor B D) (CategoryTheory.Functor B E)))

A version of CategoryTheory.whiskeringRight for bifunctors, obtained by uncurrying, applying whiskeringRight and currying back

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theorem CategoryTheory.whiskeringRight₂_obj_map_app_app (B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) : ∀ {X_1 Y : CategoryTheory.Functor B C} (f : X_1 ⟶ Y) (Y_1 : CategoryTheory.Functor B D) (X_2 : B), ((((CategoryTheory.whiskeringRight₂ B C D E).obj X).map f).app Y_1).app X_2 = (X.map (f.app X_2)).app (Y_1.obj X_2)

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theorem CategoryTheory.whiskeringRight₂_obj_obj_map_app (B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : CategoryTheory.Functor B C) : ∀ {X_1 Y : CategoryTheory.Functor B D} (g : X_1 ⟶ Y) (X_2 : B), ((((CategoryTheory.whiskeringRight₂ B C D E).obj X✝).obj X).map g).app X_2 = (X✝.obj (X.obj X_2)).map (g.app X_2)

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theorem CategoryTheory.whiskeringRight₂_obj_obj_obj_map (B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : CategoryTheory.Functor B C) (Y : CategoryTheory.Functor B D) : ∀ {X_1 Y_1 : B} (f : X_1 ⟶ Y_1), ((((CategoryTheory.whiskeringRight₂ B C D E).obj X✝).obj X).obj Y).map f = CategoryTheory.CategoryStruct.comp ((X✝.map (X.map f)).app (Y.obj X_1)) ((X✝.obj (X.obj Y_1)).map (Y.map f))

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theorem CategoryTheory.whiskeringRight₂_map_app_app_app (B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] : ∀ {X Y : CategoryTheory.Functor C (CategoryTheory.Functor D E)} (f : X ⟶ Y) (X_1 : CategoryTheory.Functor B C) (Y_1 : CategoryTheory.Functor B D) (c : B), ((((CategoryTheory.whiskeringRight₂ B C D E).map f).app X_1).app Y_1).app c = (f.app (X_1.obj c)).app (Y_1.obj c)

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theorem CategoryTheory.whiskeringRight₂_obj_obj_obj_obj (B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : CategoryTheory.Functor B C) (Y : CategoryTheory.Functor B D) (X : B) : ((((CategoryTheory.whiskeringRight₂ B C D E).obj X✝¹).obj X✝).obj Y).obj X = (X✝¹.obj (X✝.obj X)).obj (Y.obj X)

theorem CategoryTheory.Functor.uncurry_obj_curry_obj {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] (F : CategoryTheory.Functor (B × C) D) : CategoryTheory.uncurry.obj (CategoryTheory.curry.obj F) = F

theorem CategoryTheory.Functor.curry_obj_injective {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {F₁ : CategoryTheory.Functor (C × D) E} {F₂ : CategoryTheory.Functor (C × D) E} (h : CategoryTheory.curry.obj F₁ = CategoryTheory.curry.obj F₂) : F₁ = F₂

theorem CategoryTheory.Functor.curry_obj_uncurry_obj {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] (F : CategoryTheory.Functor B (CategoryTheory.Functor C D)) : CategoryTheory.curry.obj (CategoryTheory.uncurry.obj F) = F

theorem CategoryTheory.Functor.uncurry_obj_injective {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {F₁ : CategoryTheory.Functor B (CategoryTheory.Functor C D)} {F₂ : CategoryTheory.Functor B (CategoryTheory.Functor C D)} (h : CategoryTheory.uncurry.obj F₁ = CategoryTheory.uncurry.obj F₂) : F₁ = F₂

theorem CategoryTheory.Functor.flip_flip {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] (F : CategoryTheory.Functor B (CategoryTheory.Functor C D)) : F.flip.flip = F

theorem CategoryTheory.Functor.flip_injective {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {F₁ : CategoryTheory.Functor B (CategoryTheory.Functor C D)} {F₂ : CategoryTheory.Functor B (CategoryTheory.Functor C D)} (h : F₁.flip = F₂.flip) : F₁ = F₂

theorem CategoryTheory.Functor.uncurry_obj_curry_obj_flip_flip {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {H : Type u₅} [CategoryTheory.Category.{v₅, u₅} H] (F₁ : CategoryTheory.Functor B C) (F₂ : CategoryTheory.Functor D E) (G : CategoryTheory.Functor (C × E) H) : CategoryTheory.uncurry.obj (F₂.comp (F₁.comp (CategoryTheory.curry.obj G)).flip).flip = (F₁.prod F₂).comp G

theorem CategoryTheory.Functor.uncurry_obj_curry_obj_flip_flip' {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {H : Type u₅} [CategoryTheory.Category.{v₅, u₅} H] (F₁ : CategoryTheory.Functor B C) (F₂ : CategoryTheory.Functor D E) (G : CategoryTheory.Functor (C × E) H) : CategoryTheory.uncurry.obj (F₁.comp (F₂.comp (CategoryTheory.curry.obj G).flip).flip) = (F₁.prod F₂).comp G