ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Monoidal.MonFunctorCategoryEquivalence.counitIso_hom_app_app_hom
@[simps!]
def counitIso : inverse ⋙ functor ≅ 𝟭 (C ⥤ Mon_ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (X : C ⥤ Mon_ D)
Goal: ∀ (X_1 : C), ((CategoryTheory.Monoidal.MonFunctorCategoryEquivalence.counitIso.hom.app X).app X_1).hom = 𝟙 (X.obj X_1).X
-- CategoryTheory.Monoidal.MonFunctorCategoryEquivalence.counitIso_inv_app_app_hom
@[simps!]
def counitIso : inverse ⋙ functor ≅ 𝟭 (C ⥤ Mon_ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (X : C ⥤ Mon_ D)
Goal: ∀ (X_1 : C), ((CategoryTheory.Monoidal.MonFunctorCategoryEquivalence.counitIso.inv.app X).app X_1).hom = 𝟙 (X.obj X_1).X
-- CategoryTheory.Monoidal.monFunctorCategoryEquivalence_counitIso
@[simps]
def monFunctorCategoryEquivalence : Mon_ (C ⥤ D) ≌ C ⥤ Mon_ D 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D]
Goal: (CategoryTheory.Monoidal.monFunctorCategoryEquivalence C D).counitIso =
  CategoryTheory.Monoidal.MonFunctorCategoryEquivalence.counitIso
-- CategoryTheory.Monoidal.monFunctorCategoryEquivalence_unitIso
@[simps]
def monFunctorCategoryEquivalence : Mon_ (C ⥤ D) ≌ C ⥤ Mon_ D 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D]
Goal: (CategoryTheory.Monoidal.monFunctorCategoryEquivalence C D).unitIso =
  CategoryTheory.Monoidal.MonFunctorCategoryEquivalence.unitIso
-- CategoryTheory.Monoidal.monFunctorCategoryEquivalence_functor
@[simps]
def monFunctorCategoryEquivalence : Mon_ (C ⥤ D) ≌ C ⥤ Mon_ D 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D]
Goal: (CategoryTheory.Monoidal.monFunctorCategoryEquivalence C D).functor =
  CategoryTheory.Monoidal.MonFunctorCategoryEquivalence.functor
-- CategoryTheory.Monoidal.monFunctorCategoryEquivalence_inverse
@[simps]
def monFunctorCategoryEquivalence : Mon_ (C ⥤ D) ≌ C ⥤ Mon_ D 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D]
Goal: (CategoryTheory.Monoidal.monFunctorCategoryEquivalence C D).inverse =
  CategoryTheory.Monoidal.MonFunctorCategoryEquivalence.inverse
-- CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functorObj_obj_counit
@[simps]
def functorObj (A : Comon_ (C ⥤ D)) : C ⥤ Comon_ D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (A : Comon_ (C ⥤ D)) (X : C)
Goal: ((CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functorObj A).obj X).counit = A.counit.app X
-- CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functorObj_map_hom
@[simps]
def functorObj (A : Comon_ (C ⥤ D)) : C ⥤ Comon_ D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (A : Comon_ (C ⥤ D))
Goal: ∀ {X Y : C} (f : X ⟶ Y), ((CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functorObj A).map f).hom = A.X.map f
-- CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functorObj_obj_comul
@[simps]
def functorObj (A : Comon_ (C ⥤ D)) : C ⥤ Comon_ D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (A : Comon_ (C ⥤ D)) (X : C)
Goal: ((CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functorObj A).obj X).comul = A.comul.app X
-- CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functorObj_obj_X
@[simps]
def functorObj (A : Comon_ (C ⥤ D)) : C ⥤ Comon_ D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (A : Comon_ (C ⥤ D)) (X : C)
Goal: ((CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functorObj A).obj X).X = A.X.obj X
-- CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functor_map_app_hom
@[simps]
def functor : Comon_ (C ⥤ D) ⥤ C ⥤ Comon_ D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D]
Goal: ∀ {X Y : Comon_ (C ⥤ D)} (f : X ⟶ Y) (X_1 : C),
  ((CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functor.map f).app X_1).hom = f.hom.app X_1
-- CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functor_obj
@[simps]
def functor : Comon_ (C ⥤ D) ⥤ C ⥤ Comon_ D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (A : Comon_ (C ⥤ D))
Goal: CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functor.obj A =
  CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functorObj A
-- CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.inverseObj_comul_app
@[simps]
def inverseObj (F : C ⥤ Comon_ D) : Comon_ (C ⥤ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : C ⥤ Comon_ D) (X : C)
Goal: (CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.inverseObj F).comul.app X = (F.obj X).comul
-- CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.inverseObj_X
@[simps]
def inverseObj (F : C ⥤ Comon_ D) : Comon_ (C ⥤ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : C ⥤ Comon_ D)
Goal: (CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.inverseObj F).X = F ⋙ Comon_.forget D
-- CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.inverseObj_counit_app
@[simps]
def inverseObj (F : C ⥤ Comon_ D) : Comon_ (C ⥤ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : C ⥤ Comon_ D) (X : C)
Goal: (CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.inverseObj F).counit.app X = (F.obj X).counit
-- _private.Mathlib.CategoryTheory.Monoidal.Internal.FunctorCategory.0.CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.inverse_obj
@[simps]
private def inverse : (C ⥤ Comon_ D) ⥤ Comon_ (C ⥤ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (F : C ⥤ Comon_ D)
Goal: CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.inverse.obj F =
  CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.inverseObj F
-- _private.Mathlib.CategoryTheory.Monoidal.Internal.FunctorCategory.0.CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.inverse_map_hom_app
@[simps]
private def inverse : (C ⥤ Comon_ D) ⥤ Comon_ (C ⥤ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D]
Goal: ∀ {X Y : C ⥤ Comon_ D} (α : X ⟶ Y) (X_1 : C),
  (CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.inverse.map α).hom.app X_1 = (α.app X_1).hom
-- _private.Mathlib.CategoryTheory.Monoidal.Internal.FunctorCategory.0.CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.unitIso_inv_app_hom_app
@[simps!]
private def unitIso : 𝟭 (Comon_ (C ⥤ D)) ≅ functor ⋙ inverse 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (X : Comon_ (C ⥤ D))
Goal: ∀ (x : C), (CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.unitIso.inv.app X).hom.app x = 𝟙 (X.X.obj x)
-- _private.Mathlib.CategoryTheory.Monoidal.Internal.FunctorCategory.0.CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.unitIso_hom_app_hom_app
@[simps!]
private def unitIso : 𝟭 (Comon_ (C ⥤ D)) ≅ functor ⋙ inverse 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (X : Comon_ (C ⥤ D))
Goal: ∀ (x : C), (CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.unitIso.hom.app X).hom.app x = 𝟙 (X.X.obj x)
-- CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.counitIso_hom_app_app_hom
@[simps!]
def counitIso : inverse ⋙ functor ≅ 𝟭 (C ⥤ Comon_ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (X : C ⥤ Comon_ D)
Goal: ∀ (X_1 : C),
  ((CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.counitIso.hom.app X).app X_1).hom = 𝟙 (X.obj X_1).X
-- CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.counitIso_inv_app_app_hom
@[simps!]
def counitIso : inverse ⋙ functor ≅ 𝟭 (C ⥤ Comon_ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] (X : C ⥤ Comon_ D)
Goal: ∀ (X_1 : C),
  ((CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.counitIso.inv.app X).app X_1).hom = 𝟙 (X.obj X_1).X
-- CategoryTheory.Monoidal.comonFunctorCategoryEquivalence_inverse
@[simps]
def comonFunctorCategoryEquivalence : Comon_ (C ⥤ D) ≌ C ⥤ Comon_ D 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D]
Goal: (CategoryTheory.Monoidal.comonFunctorCategoryEquivalence C D).inverse =
  CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.inverse
-- CategoryTheory.Monoidal.comonFunctorCategoryEquivalence_functor
@[simps]
def comonFunctorCategoryEquivalence : Comon_ (C ⥤ D) ≌ C ⥤ Comon_ D 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D]
Goal: (CategoryTheory.Monoidal.comonFunctorCategoryEquivalence C D).functor =
  CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functor
-- CategoryTheory.Monoidal.comonFunctorCategoryEquivalence_counitIso
@[simps]
def comonFunctorCategoryEquivalence : Comon_ (C ⥤ D) ≌ C ⥤ Comon_ D 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D]
Goal: (CategoryTheory.Monoidal.comonFunctorCategoryEquivalence C D).counitIso =
  CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.counitIso
-- CategoryTheory.Monoidal.comonFunctorCategoryEquivalence_unitIso
@[simps]
def comonFunctorCategoryEquivalence : Comon_ (C ⥤ D) ≌ C ⥤ Comon_ D 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D]
Goal: (CategoryTheory.Monoidal.comonFunctorCategoryEquivalence C D).unitIso =
  CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.unitIso
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor_map_app_hom
@[simps!]
def functor : CommMon_ (C ⥤ D) ⥤ C ⥤ CommMon_ D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D]
Goal: ∀ {X Y : CommMon_ (C ⥤ D)} (f : X ⟶ Y) (X_1 : C),
  ((CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor.map f).app X_1).hom = f.hom.app X_1
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor_obj_map_hom
@[simps!]
def functor : CommMon_ (C ⥤ D) ⥤ C ⥤ CommMon_ D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (A : CommMon_ (C ⥤ D))
Goal: ∀ {X Y : C} (a : X ⟶ Y),
  ((CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor.obj A).map a).hom = A.X.map a
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor_obj_obj_mul
@[simps!]
def functor : CommMon_ (C ⥤ D) ⥤ C ⥤ CommMon_ D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (A : CommMon_ (C ⥤ D)) (X : C)
Goal: ((CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor.obj A).obj X).mul = A.mul.app X
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor_obj_obj_one
@[simps!]
def functor : CommMon_ (C ⥤ D) ⥤ C ⥤ CommMon_ D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (A : CommMon_ (C ⥤ D)) (X : C)
Goal: ((CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor.obj A).obj X).one = A.one.app X
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor_obj_obj_X
@[simps!]
def functor : CommMon_ (C ⥤ D) ⥤ C ⥤ CommMon_ D 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (A : CommMon_ (C ⥤ D)) (X : C)
Goal: ((CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor.obj A).obj X).X = A.X.obj X
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.inverse_map_hom_app
@[simps!]
def inverse : (C ⥤ CommMon_ D) ⥤ CommMon_ (C ⥤ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D]
Goal: ∀ {X Y : C ⥤ CommMon_ D} (α : X ⟶ Y) (X_1 : C),
  (CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.inverse.map α).hom.app X_1 = (α.app X_1).hom
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.inverse_obj_mul_app
@[simps!]
def inverse : (C ⥤ CommMon_ D) ⥤ CommMon_ (C ⥤ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (F : C ⥤ CommMon_ D) (X : C)
Goal: (CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.inverse.obj F).mul.app X = (F.obj X).mul
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.inverse_obj_X_obj
@[simps!]
def inverse : (C ⥤ CommMon_ D) ⥤ CommMon_ (C ⥤ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (F : C ⥤ CommMon_ D) (X : C)
Goal: (CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.inverse.obj F).X.obj X =
  ((CommMon_.forget₂Mon_ D).obj (F.obj X)).X
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.inverse_obj_X_map
@[simps!]
def inverse : (C ⥤ CommMon_ D) ⥤ CommMon_ (C ⥤ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (F : C ⥤ CommMon_ D)
Goal: ∀ {X Y : C} (f : X ⟶ Y),
  (CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.inverse.obj F).X.map f = (F.map f).hom
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.inverse_obj_one_app
@[simps!]
def inverse : (C ⥤ CommMon_ D) ⥤ CommMon_ (C ⥤ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (F : C ⥤ CommMon_ D) (X : C)
Goal: (CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.inverse.obj F).one.app X = (F.obj X).one
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.unitIso_inv_app_hom_app
@[simps!]
def unitIso : 𝟭 (CommMon_ (C ⥤ D)) ≅ functor ⋙ inverse 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (X : CommMon_ (C ⥤ D))
Goal: ∀ (x : C),
  (CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.unitIso.inv.app X).hom.app x =
    𝟙 ((CommMon_.forget₂Mon_ D).obj ((CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor.obj X).obj x)).X
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.unitIso_hom_app_hom_app
@[simps!]
def unitIso : 𝟭 (CommMon_ (C ⥤ D)) ≅ functor ⋙ inverse 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (X : CommMon_ (C ⥤ D))
Goal: ∀ (x : C), (CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.unitIso.hom.app X).hom.app x = 𝟙 (X.X.obj x)
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.counitIso_hom_app_app_hom
@[simps!]
def counitIso : inverse ⋙ functor ≅ 𝟭 (C ⥤ CommMon_ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (X : C ⥤ CommMon_ D)
Goal: ∀ (X_1 : C),
  ((CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.counitIso.hom.app X).app X_1).hom =
    𝟙 ((CommMon_.forget₂Mon_ D).obj (X.obj X_1)).X
-- CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.counitIso_inv_app_app_hom
@[simps!]
def counitIso : inverse ⋙ functor ≅ 𝟭 (C ⥤ CommMon_ D) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D] (X : C ⥤ CommMon_ D)
Goal: ∀ (X_1 : C),
  ((CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.counitIso.inv.app X).app X_1).hom = 𝟙 (X.obj X_1).X
-- CategoryTheory.Monoidal.commMonFunctorCategoryEquivalence_unitIso
@[simps]
def commMonFunctorCategoryEquivalence : CommMon_ (C ⥤ D) ≌ C ⥤ CommMon_ D 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D]
Goal: (CategoryTheory.Monoidal.commMonFunctorCategoryEquivalence C D).unitIso =
  CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.unitIso
-- CategoryTheory.Monoidal.commMonFunctorCategoryEquivalence_counitIso
@[simps]
def commMonFunctorCategoryEquivalence : CommMon_ (C ⥤ D) ≌ C ⥤ CommMon_ D 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D]
Goal: (CategoryTheory.Monoidal.commMonFunctorCategoryEquivalence C D).counitIso =
  CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.counitIso
-- CategoryTheory.Monoidal.commMonFunctorCategoryEquivalence_inverse
@[simps]
def commMonFunctorCategoryEquivalence : CommMon_ (C ⥤ D) ≌ C ⥤ CommMon_ D 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D]
Goal: (CategoryTheory.Monoidal.commMonFunctorCategoryEquivalence C D).inverse =
  CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.inverse
-- CategoryTheory.Monoidal.commMonFunctorCategoryEquivalence_functor
@[simps]
def commMonFunctorCategoryEquivalence : CommMon_ (C ⥤ D) ≌ C ⥤ CommMon_ D 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.MonoidalCategory D] [CategoryTheory.BraidedCategory D]
Goal: (CategoryTheory.Monoidal.commMonFunctorCategoryEquivalence C D).functor =
  CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor
-- Polynomial.cardinal_mk_eq_max
@[simp]
theorem cardinal_mk_eq_max {R : Type u} [Semiring R] [Nontrivial R] : #(R[X]) = max #R ℵ₀ 
Arguments: {R : Type u} [Semiring R] [Nontrivial R]
Goal: #R[X] = max #R ℵ₀
-- Polynomial.cardinal_mk_le_max
theorem cardinal_mk_le_max {R : Type u} [Semiring R] : #(R[X]) ≤ max #R ℵ₀ 
Arguments: {R : Type u} [Semiring R]
Goal: #R[X] ≤ max #R ℵ₀
-- CategoryTheory.RepresentablyFlat.cofiltered
class RepresentablyFlat (F : C ⥤ D) : Prop where
  cofiltered : ∀ X : D, IsCofiltered (StructuredArrow X F)
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : C ⥤ D} [self : CategoryTheory.RepresentablyFlat F] (X : D)
Goal: CategoryTheory.IsCofiltered (CategoryTheory.StructuredArrow X F)
-- CategoryTheory.RepresentablyFlat.id
theorem RepresentablyFlat.id : RepresentablyFlat (𝟭 C) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C]
Goal: CategoryTheory.RepresentablyFlat (𝟭 C)
-- CategoryTheory.flat_of_preservesFiniteLimits
theorem flat_of_preservesFiniteLimits [HasFiniteLimits C] (F : C ⥤ D) [PreservesFiniteLimits F] :
    RepresentablyFlat F 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Limits.HasFiniteLimits C] (F : C ⥤ D) [CategoryTheory.Limits.PreservesFiniteLimits F]
Goal: CategoryTheory.RepresentablyFlat F
-- CategoryTheory.PreservesFiniteLimitsOfFlat.fac
theorem fac (x : J) : lift F hc s ≫ (F.mapCone c).π.app x = s.π.app x 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {J : Type v₁} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] {K : J ⥤ C} (F : C ⥤ D) [CategoryTheory.RepresentablyFlat F] {c : CategoryTheory.Limits.Cone K} (hc : CategoryTheory.Limits.IsLimit c) (s : CategoryTheory.Limits.Cone (K ⋙ F)) (x : J)
Goal: CategoryTheory.PreservesFiniteLimitsOfFlat.lift F hc s ≫ (F.mapCone c).π.app x = s.π.app x
-- CategoryTheory.PreservesFiniteLimitsOfFlat.uniq
theorem uniq {K : J ⥤ C} {c : Cone K} (hc : IsLimit c) (s : Cone (K ⋙ F))
    (f₁ f₂ : s.pt ⟶ F.obj c.pt) (h₁ : ∀ j : J, f₁ ≫ (F.mapCone c).π.app j = s.π.app j)
    (h₂ : ∀ j : J, f₂ ≫ (F.mapCone c).π.app j = s.π.app j) : f₁ = f₂ 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {J : Type v₁} [CategoryTheory.SmallCategory J] [CategoryTheory.FinCategory J] (F : C ⥤ D) [CategoryTheory.RepresentablyFlat F] {K : J ⥤ C} {c : CategoryTheory.Limits.Cone K} (hc : CategoryTheory.Limits.IsLimit c) (s : CategoryTheory.Limits.Cone (K ⋙ F)) (f₁ f₂ : s.pt ⟶ F.obj c.pt) (h₁ : ∀ (j : J), f₁ ≫ (F.mapCone c).π.app j = s.π.app j) (h₂ : ∀ (j : J), f₂ ≫ (F.mapCone c).π.app j = s.π.app j)
Goal: f₁ = f₂
-- CategoryTheory.flat_iff_lan_flat
theorem flat_iff_lan_flat (F : C ⥤ D) :
    RepresentablyFlat F ↔ RepresentablyFlat (F.op.lan : _ ⥤ Dᵒᵖ ⥤ Type u₁) 
Arguments: {C D : Type u₁} [CategoryTheory.SmallCategory C] [CategoryTheory.SmallCategory D] [CategoryTheory.Limits.HasFiniteLimits C] (F : C ⥤ D)
Goal: CategoryTheory.RepresentablyFlat F ↔ CategoryTheory.RepresentablyFlat F.op.lan
-- Module.AEval.isTorsion_of_aeval_eq_zero
theorem isTorsion_of_aeval_eq_zero [CommSemiring R] [NoZeroDivisors R] [Semiring A] [Algebra R A]
    [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M]
    {p : R[X]} (h : aeval a p = 0) (h' : p ≠ 0) :
    IsTorsion R[X] (AEval R M a) 
Arguments: {R : Type u_1} {M : Type u_3} {A : Type u_4} {a : A} [CommSemiring R] [NoZeroDivisors R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] {p : R[X]} (h : (Polynomial.aeval a) p = 0) (h' : p ≠ 0)
Goal: Module.IsTorsion R[X] (Module.AEval R M a)
-- Module.AEval.isTorsion_of_finiteDimensional
theorem isTorsion_of_finiteDimensional [Field K] [Ring A] [Algebra K A]
    [AddCommGroup M] [Module A M] [Module K M] [IsScalarTower K A M] [FiniteDimensional K A] :
    IsTorsion K[X] (AEval K M a) 
Arguments: (K : Type u_2) (M : Type u_3) {A : Type u_4} (a : A) [Field K] [Ring A] [Algebra K A] [AddCommGroup M] [Module A M] [Module K M] [IsScalarTower K A M] [FiniteDimensional K A]
Goal: Module.IsTorsion K[X] (Module.AEval K M a)