ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Monoidal.transportedMonoidalCounitIso_hom
@[simps! hom inv]
def transportedMonoidalCounitIso (e : C ≌ D) :
    (fromTransported e).toLaxMonoidalFunctor ⊗⋙ (toTransported e).toLaxMonoidalFunctor ≅
      LaxMonoidalFunctor.id (Transported e) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (e : C ≌ D)
Goal: (CategoryTheory.Monoidal.transportedMonoidalCounitIso e).hom =
  CategoryTheory.inv (CategoryTheory.monoidalUnit (CategoryTheory.Monoidal.fromTransported e) e.symm.toAdjunction)
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctor_map_hom₃
@[simps]
noncomputable def Triangle.shiftFunctor (n : ℤ) : Triangle C ⥤ Triangle C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] (n : ℤ)
Goal: ∀ {X Y : CategoryTheory.Pretriangulated.Triangle C} (f : X ⟶ Y),
  ((CategoryTheory.Pretriangulated.Triangle.shiftFunctor C n).map f).hom₃ = (CategoryTheory.shiftFunctor C n).map f.hom₃
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctor_map_hom₁
@[simps]
noncomputable def Triangle.shiftFunctor (n : ℤ) : Triangle C ⥤ Triangle C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] (n : ℤ)
Goal: ∀ {X Y : CategoryTheory.Pretriangulated.Triangle C} (f : X ⟶ Y),
  ((CategoryTheory.Pretriangulated.Triangle.shiftFunctor C n).map f).hom₁ = (CategoryTheory.shiftFunctor C n).map f.hom₁
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctor_obj
@[simps]
noncomputable def Triangle.shiftFunctor (n : ℤ) : Triangle C ⥤ Triangle C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] (n : ℤ) (T : CategoryTheory.Pretriangulated.Triangle C)
Goal: (CategoryTheory.Pretriangulated.Triangle.shiftFunctor C n).obj T =
  CategoryTheory.Pretriangulated.Triangle.mk (n.negOnePow • (CategoryTheory.shiftFunctor C n).map T.mor₁)
    (n.negOnePow • (CategoryTheory.shiftFunctor C n).map T.mor₂)
    (n.negOnePow •
      CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctor C n).map T.mor₃)
        ((CategoryTheory.shiftFunctorComm C 1 n).hom.app T.obj₁))
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctor_map_hom₂
@[simps]
noncomputable def Triangle.shiftFunctor (n : ℤ) : Triangle C ⥤ Triangle C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] (n : ℤ)
Goal: ∀ {X Y : CategoryTheory.Pretriangulated.Triangle C} (f : X ⟶ Y),
  ((CategoryTheory.Pretriangulated.Triangle.shiftFunctor C n).map f).hom₂ = (CategoryTheory.shiftFunctor C n).map f.hom₂
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero_inv_app_hom₁
@[simps!]
noncomputable def Triangle.shiftFunctorZero : Triangle.shiftFunctor C 0 ≅ 𝟭 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero C).inv.app X).hom₁ =
  (CategoryTheory.shiftFunctorZero C ℤ).inv.app X.obj₁
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero_inv_app_hom₃
@[simps!]
noncomputable def Triangle.shiftFunctorZero : Triangle.shiftFunctor C 0 ≅ 𝟭 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero C).inv.app X).hom₃ =
  (CategoryTheory.shiftFunctorZero C ℤ).inv.app X.obj₃
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero_hom_app_hom₁
@[simps!]
noncomputable def Triangle.shiftFunctorZero : Triangle.shiftFunctor C 0 ≅ 𝟭 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero C).hom.app X).hom₁ =
  (CategoryTheory.shiftFunctorZero C ℤ).hom.app X.obj₁
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero_hom_app_hom₂
@[simps!]
noncomputable def Triangle.shiftFunctorZero : Triangle.shiftFunctor C 0 ≅ 𝟭 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero C).hom.app X).hom₂ =
  (CategoryTheory.shiftFunctorZero C ℤ).hom.app X.obj₂
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero_hom_app_hom₃
@[simps!]
noncomputable def Triangle.shiftFunctorZero : Triangle.shiftFunctor C 0 ≅ 𝟭 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero C).hom.app X).hom₃ =
  (CategoryTheory.shiftFunctorZero C ℤ).hom.app X.obj₃
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero_inv_app_hom₂
@[simps!]
noncomputable def Triangle.shiftFunctorZero : Triangle.shiftFunctor C 0 ≅ 𝟭 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero C).inv.app X).hom₂ =
  (CategoryTheory.shiftFunctorZero C ℤ).inv.app X.obj₂
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd'_inv_app_hom₂
@[simps!]
noncomputable def Triangle.shiftFunctorAdd' (a b n : ℤ) (h : a + b = n) :
    Triangle.shiftFunctor C n ≅ Triangle.shiftFunctor C a ⋙ Triangle.shiftFunctor C b 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (a b n : ℤ) (h : a + b = n) (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd' C a b n h).inv.app X).hom₂ =
  (CategoryTheory.shiftFunctorAdd' C a b n h).inv.app X.obj₂
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd'_hom_app_hom₂
@[simps!]
noncomputable def Triangle.shiftFunctorAdd' (a b n : ℤ) (h : a + b = n) :
    Triangle.shiftFunctor C n ≅ Triangle.shiftFunctor C a ⋙ Triangle.shiftFunctor C b 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (a b n : ℤ) (h : a + b = n) (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd' C a b n h).hom.app X).hom₂ =
  (CategoryTheory.shiftFunctorAdd' C a b n h).hom.app X.obj₂
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd'_hom_app_hom₁
@[simps!]
noncomputable def Triangle.shiftFunctorAdd' (a b n : ℤ) (h : a + b = n) :
    Triangle.shiftFunctor C n ≅ Triangle.shiftFunctor C a ⋙ Triangle.shiftFunctor C b 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (a b n : ℤ) (h : a + b = n) (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd' C a b n h).hom.app X).hom₁ =
  (CategoryTheory.shiftFunctorAdd' C a b n h).hom.app X.obj₁
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd'_inv_app_hom₃
@[simps!]
noncomputable def Triangle.shiftFunctorAdd' (a b n : ℤ) (h : a + b = n) :
    Triangle.shiftFunctor C n ≅ Triangle.shiftFunctor C a ⋙ Triangle.shiftFunctor C b 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (a b n : ℤ) (h : a + b = n) (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd' C a b n h).inv.app X).hom₃ =
  (CategoryTheory.shiftFunctorAdd' C a b n h).inv.app X.obj₃
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd'_hom_app_hom₃
@[simps!]
noncomputable def Triangle.shiftFunctorAdd' (a b n : ℤ) (h : a + b = n) :
    Triangle.shiftFunctor C n ≅ Triangle.shiftFunctor C a ⋙ Triangle.shiftFunctor C b 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (a b n : ℤ) (h : a + b = n) (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd' C a b n h).hom.app X).hom₃ =
  (CategoryTheory.shiftFunctorAdd' C a b n h).hom.app X.obj₃
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd'_inv_app_hom₁
@[simps!]
noncomputable def Triangle.shiftFunctorAdd' (a b n : ℤ) (h : a + b = n) :
    Triangle.shiftFunctor C n ≅ Triangle.shiftFunctor C a ⋙ Triangle.shiftFunctor C b 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (a b n : ℤ) (h : a + b = n) (X : CategoryTheory.Pretriangulated.Triangle C)
Goal: ((CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd' C a b n h).inv.app X).hom₁ =
  (CategoryTheory.shiftFunctorAdd' C a b n h).inv.app X.obj₁
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctor_eq
@[simp]
lemma shiftFunctor_eq (n : ℤ) :
    CategoryTheory.shiftFunctor (Triangle C) n = Triangle.shiftFunctor C n 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (n : ℤ)
Goal: CategoryTheory.shiftFunctor (CategoryTheory.Pretriangulated.Triangle C) n =
  CategoryTheory.Pretriangulated.Triangle.shiftFunctor C n
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero_eq
@[simp]
lemma shiftFunctorZero_eq :
    CategoryTheory.shiftFunctorZero (Triangle C) ℤ = Triangle.shiftFunctorZero C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive]
Goal: CategoryTheory.shiftFunctorZero (CategoryTheory.Pretriangulated.Triangle C) ℤ =
  CategoryTheory.Pretriangulated.Triangle.shiftFunctorZero C
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd_eq
@[simp]
lemma shiftFunctorAdd_eq (a b : ℤ) :
    CategoryTheory.shiftFunctorAdd (Triangle C) a b =
      Triangle.shiftFunctorAdd' C a b _ rfl 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (a b : ℤ)
Goal: CategoryTheory.shiftFunctorAdd (CategoryTheory.Pretriangulated.Triangle C) a b =
  CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd' C a b (a + b) ⋯
-- CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd'_eq
@[simp]
lemma shiftFunctorAdd'_eq (a b c : ℤ) (h : a + b = c) :
    CategoryTheory.shiftFunctorAdd' (Triangle C) a b c h =
      Triangle.shiftFunctorAdd' C a b c h 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.HasShift C ℤ] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (a b c : ℤ) (h : a + b = c)
Goal: CategoryTheory.shiftFunctorAdd' (CategoryTheory.Pretriangulated.Triangle C) a b c h =
  CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd' C a b c h
-- CategoryTheory.endofunctorMonoidalCategory_tensorUnit_obj
@[simp] theorem endofunctorMonoidalCategory_tensorUnit_obj (X : C) :
    (𝟙_ (C ⥤ C)).obj X = X 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : C)
Goal: (𝟙_ (CategoryTheory.Functor C C)).obj X = X
-- CategoryTheory.endofunctorMonoidalCategory_tensorUnit_map
@[simp] theorem endofunctorMonoidalCategory_tensorUnit_map {X Y : C} (f : X ⟶ Y) :
    (𝟙_ (C ⥤ C)).map f = f 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y)
Goal: (𝟙_ (CategoryTheory.Functor C C)).map f = f
-- CategoryTheory.endofunctorMonoidalCategory_tensorObj_obj
@[simp] theorem endofunctorMonoidalCategory_tensorObj_obj (F G : C ⥤ C) (X : C) :
    (F ⊗ G).obj X = G.obj (F.obj X) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F G : CategoryTheory.Functor C C) (X : C)
Goal: (CategoryTheory.MonoidalCategory.tensorObj F G).obj X = G.obj (F.obj X)
-- CategoryTheory.endofunctorMonoidalCategory_tensorObj_map
@[simp] theorem endofunctorMonoidalCategory_tensorObj_map (F G : C ⥤ C) {X Y : C} (f : X ⟶ Y) :
    (F ⊗ G).map f = G.map (F.map f) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F G : CategoryTheory.Functor C C) {X Y : C} (f : X ⟶ Y)
Goal: (CategoryTheory.MonoidalCategory.tensorObj F G).map f = G.map (F.map f)
-- CategoryTheory.endofunctorMonoidalCategory_tensorMap_app
@[simp] theorem endofunctorMonoidalCategory_tensorMap_app
    {F G H K : C ⥤ C} {α : F ⟶ G} {β : H ⟶ K} (X : C) :
    (α ⊗ β).app X = β.app (F.obj X) ≫ K.map (α.app X) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] {F G H K : CategoryTheory.Functor C C} {α : F ⟶ G} {β : H ⟶ K} (X : C)
Goal: (CategoryTheory.MonoidalCategory.tensorHom α β).app X =
  CategoryTheory.CategoryStruct.comp (β.app (F.obj X)) (K.map (α.app X))
-- CategoryTheory.endofunctorMonoidalCategory_whiskerLeft_app
@[simp] theorem endofunctorMonoidalCategory_whiskerLeft_app
    {F H K : C ⥤ C} {β : H ⟶ K} (X : C) :
    (F ◁ β).app X = β.app (F.obj X) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] {F H K : CategoryTheory.Functor C C} {β : H ⟶ K} (X : C)
Goal: (CategoryTheory.MonoidalCategory.whiskerLeft F β).app X = β.app (F.obj X)
-- CategoryTheory.endofunctorMonoidalCategory_whiskerRight_app
@[simp] theorem endofunctorMonoidalCategory_whiskerRight_app
    {F G H : C ⥤ C} {α : F ⟶ G} (X : C) :
    (α ▷ H).app X = H.map (α.app X) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] {F G H : CategoryTheory.Functor C C} {α : F ⟶ G} (X : C)
Goal: (CategoryTheory.MonoidalCategory.whiskerRight α H).app X = H.map (α.app X)
-- CategoryTheory.endofunctorMonoidalCategory_associator_hom_app
@[simp] theorem endofunctorMonoidalCategory_associator_hom_app (F G H : C ⥤ C) (X : C) :
  (α_ F G H).hom.app X = 𝟙 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F G H : CategoryTheory.Functor C C) (X : C)
Goal: (CategoryTheory.MonoidalCategory.associator F G H).hom.app X =
  CategoryTheory.CategoryStruct.id
    ((CategoryTheory.MonoidalCategory.tensorObj (CategoryTheory.MonoidalCategory.tensorObj F G) H).obj X)
-- CategoryTheory.endofunctorMonoidalCategory_associator_inv_app
@[simp] theorem endofunctorMonoidalCategory_associator_inv_app (F G H : C ⥤ C) (X : C) :
  (α_ F G H).inv.app X = 𝟙 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F G H : CategoryTheory.Functor C C) (X : C)
Goal: (CategoryTheory.MonoidalCategory.associator F G H).inv.app X =
  CategoryTheory.CategoryStruct.id
    ((CategoryTheory.MonoidalCategory.tensorObj F (CategoryTheory.MonoidalCategory.tensorObj G H)).obj X)
-- CategoryTheory.endofunctorMonoidalCategory_leftUnitor_hom_app
@[simp] theorem endofunctorMonoidalCategory_leftUnitor_hom_app (F : C ⥤ C) (X : C) :
  (λ_ F).hom.app X = 𝟙 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor C C) (X : C)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor F).hom.app X =
  CategoryTheory.CategoryStruct.id
    ((CategoryTheory.MonoidalCategory.tensorObj (𝟙_ (CategoryTheory.Functor C C)) F).obj X)
-- CategoryTheory.endofunctorMonoidalCategory_leftUnitor_inv_app
@[simp] theorem endofunctorMonoidalCategory_leftUnitor_inv_app (F : C ⥤ C) (X : C) :
  (λ_ F).inv.app X = 𝟙 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor C C) (X : C)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor F).inv.app X = CategoryTheory.CategoryStruct.id (F.obj X)
-- CategoryTheory.endofunctorMonoidalCategory_rightUnitor_hom_app
@[simp] theorem endofunctorMonoidalCategory_rightUnitor_hom_app (F : C ⥤ C) (X : C) :
  (ρ_ F).hom.app X = 𝟙 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor C C) (X : C)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor F).hom.app X =
  CategoryTheory.CategoryStruct.id
    ((CategoryTheory.MonoidalCategory.tensorObj F (𝟙_ (CategoryTheory.Functor C C))).obj X)
-- CategoryTheory.endofunctorMonoidalCategory_rightUnitor_inv_app
@[simp] theorem endofunctorMonoidalCategory_rightUnitor_inv_app (F : C ⥤ C) (X : C) :
  (ρ_ F).inv.app X = 𝟙 _ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor C C) (X : C)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor F).inv.app X = CategoryTheory.CategoryStruct.id (F.obj X)
-- CategoryTheory.tensoringRightMonoidal_toLaxMonoidalFunctor_μ_app
@[simps!]
def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] (X Y : C)
Goal: ∀ (X_1 : C),
  ((CategoryTheory.tensoringRightMonoidal C).μ X Y).app X_1 = (CategoryTheory.MonoidalCategory.associator X_1 X Y).hom
-- CategoryTheory.tensoringRightMonoidal_toLaxMonoidalFunctor_ε_app
@[simps!]
def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.tensoringRightMonoidal C).ε.app X = (CategoryTheory.MonoidalCategory.rightUnitor X).inv
-- CategoryTheory.tensoringRightMonoidal_toLaxMonoidalFunctor_toFunctor_obj_obj
@[simps!]
def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] (k j : C)
Goal: ((CategoryTheory.tensoringRightMonoidal C).obj k).obj j = CategoryTheory.MonoidalCategory.tensorObj j k
-- CategoryTheory.tensoringRightMonoidal_toLaxMonoidalFunctor_toFunctor_obj_map
@[simps!]
def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C] (k : C)
Goal: ∀ {X Y : C} (f : X ⟶ Y),
  ((CategoryTheory.tensoringRightMonoidal C).obj k).map f = CategoryTheory.MonoidalCategory.whiskerRight f k
-- CategoryTheory.tensoringRightMonoidal_toLaxMonoidalFunctor_toFunctor_map_app
@[simps!]
def tensoringRightMonoidal [MonoidalCategory.{v} C] : MonoidalFunctor C (C ⥤ C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.MonoidalCategory C]
Goal: ∀ {X Y : C} (f : X ⟶ Y) (j : C),
  ((CategoryTheory.tensoringRightMonoidal C).map f).app j = CategoryTheory.MonoidalCategory.whiskerLeft j f
-- CategoryTheory.μ_hom_inv_app
@[reassoc (attr := simp)]
theorem μ_hom_inv_app (i j : M) (X : C) : (F.μ i j).app X ≫ (F.μIso i j).inv.app X = 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {M : Type u_1} [CategoryTheory.Category.{u_2, u_1} M] [CategoryTheory.MonoidalCategory M] (F : CategoryTheory.MonoidalFunctor M (CategoryTheory.Functor C C)) (i j : M) (X : C)
Goal: CategoryTheory.CategoryStruct.comp ((F.μ i j).app X) ((F.μIso i j).inv.app X) =
  CategoryTheory.CategoryStruct.id ((CategoryTheory.MonoidalCategory.tensorObj (F.obj i) (F.obj j)).obj X)
-- CategoryTheory.μ_inv_hom_app
@[reassoc (attr := simp)]
theorem μ_inv_hom_app (i j : M) (X : C) : (F.μIso i j).inv.app X ≫ (F.μ i j).app X = 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {M : Type u_1} [CategoryTheory.Category.{u_2, u_1} M] [CategoryTheory.MonoidalCategory M] (F : CategoryTheory.MonoidalFunctor M (CategoryTheory.Functor C C)) (i j : M) (X : C)
Goal: CategoryTheory.CategoryStruct.comp ((F.μIso i j).inv.app X) ((F.μ i j).app X) =
  CategoryTheory.CategoryStruct.id ((F.obj (CategoryTheory.MonoidalCategory.tensorObj i j)).obj X)
-- CategoryTheory.ε_hom_inv_app
@[reassoc (attr := simp)]
theorem ε_hom_inv_app (X : C) : F.ε.app X ≫ F.εIso.inv.app X = 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {M : Type u_1} [CategoryTheory.Category.{u_2, u_1} M] [CategoryTheory.MonoidalCategory M] (F : CategoryTheory.MonoidalFunctor M (CategoryTheory.Functor C C)) (X : C)
Goal: CategoryTheory.CategoryStruct.comp (F.ε.app X) (F.εIso.inv.app X) =
  CategoryTheory.CategoryStruct.id ((𝟙_ (CategoryTheory.Functor C C)).obj X)
-- CategoryTheory.ε_inv_hom_app
@[reassoc (attr := simp)]
theorem ε_inv_hom_app (X : C) : F.εIso.inv.app X ≫ F.ε.app X = 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {M : Type u_1} [CategoryTheory.Category.{u_2, u_1} M] [CategoryTheory.MonoidalCategory M] (F : CategoryTheory.MonoidalFunctor M (CategoryTheory.Functor C C)) (X : C)
Goal: CategoryTheory.CategoryStruct.comp (F.εIso.inv.app X) (F.ε.app X) =
  CategoryTheory.CategoryStruct.id ((F.obj (𝟙_ M)).obj X)
-- CategoryTheory.ε_naturality
@[reassoc (attr := simp)]
theorem ε_naturality {X Y : C} (f : X ⟶ Y) : F.ε.app X ≫ (F.obj (𝟙_ M)).map f = f ≫ F.ε.app Y 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {M : Type u_1} [CategoryTheory.Category.{u_2, u_1} M] [CategoryTheory.MonoidalCategory M] (F : CategoryTheory.MonoidalFunctor M (CategoryTheory.Functor C C)) {X Y : C} (f : X ⟶ Y)
Goal: CategoryTheory.CategoryStruct.comp (F.ε.app X) ((F.obj (𝟙_ M)).map f) = CategoryTheory.CategoryStruct.comp f (F.ε.app Y)
-- CategoryTheory.ε_inv_naturality
@[reassoc (attr := simp)]
theorem ε_inv_naturality {X Y : C} (f : X ⟶ Y) :
    (MonoidalFunctor.εIso F).inv.app X ≫ (𝟙_ (C ⥤ C)).map f = F.εIso.inv.app X ≫ f 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {M : Type u_1} [CategoryTheory.Category.{u_2, u_1} M] [CategoryTheory.MonoidalCategory M] (F : CategoryTheory.MonoidalFunctor M (CategoryTheory.Functor C C)) {X Y : C} (f : X ⟶ Y)
Goal: CategoryTheory.CategoryStruct.comp (F.εIso.inv.app X) ((𝟙_ (CategoryTheory.Functor C C)).map f) =
  CategoryTheory.CategoryStruct.comp (F.εIso.inv.app X) f
-- CategoryTheory.μ_naturality
@[reassoc (attr := simp)]
theorem μ_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
    (F.obj n).map ((F.obj m).map f) ≫ (F.μ m n).app Y = (F.μ m n).app X ≫ (F.obj _).map f 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {M : Type u_1} [CategoryTheory.Category.{u_2, u_1} M] [CategoryTheory.MonoidalCategory M] (F : CategoryTheory.MonoidalFunctor M (CategoryTheory.Functor C C)) {m n : M} {X Y : C} (f : X ⟶ Y)
Goal: CategoryTheory.CategoryStruct.comp ((F.obj n).map ((F.obj m).map f)) ((F.μ m n).app Y) =
  CategoryTheory.CategoryStruct.comp ((F.μ m n).app X) ((F.obj (CategoryTheory.MonoidalCategory.tensorObj m n)).map f)
-- CategoryTheory.μ_inv_naturality
@[reassoc]
theorem μ_inv_naturality {m n : M} {X Y : C} (f : X ⟶ Y) :
    (F.μIso m n).inv.app X ≫ (F.obj n).map ((F.obj m).map f) =
      (F.obj _).map f ≫ (F.μIso m n).inv.app Y 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {M : Type u_1} [CategoryTheory.Category.{u_2, u_1} M] [CategoryTheory.MonoidalCategory M] (F : CategoryTheory.MonoidalFunctor M (CategoryTheory.Functor C C)) {m n : M} {X Y : C} (f : X ⟶ Y)
Goal: CategoryTheory.CategoryStruct.comp ((F.μIso m n).inv.app X) ((F.obj n).map ((F.obj m).map f)) =
  CategoryTheory.CategoryStruct.comp ((F.obj (CategoryTheory.MonoidalCategory.tensorObj m n)).map f)
    ((F.μIso m n).inv.app Y)
-- CategoryTheory.μ_naturality₂
@[reassoc]
theorem μ_naturality₂ {m n m' n' : M} (f : m ⟶ m') (g : n ⟶ n') (X : C) :
    (F.map g).app ((F.obj m).obj X) ≫ (F.obj n').map ((F.map f).app X) ≫ (F.μ m' n').app X =
      (F.μ m n).app X ≫ (F.map (f ⊗ g)).app X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {M : Type u_1} [CategoryTheory.Category.{u_2, u_1} M] [CategoryTheory.MonoidalCategory M] (F : CategoryTheory.MonoidalFunctor M (CategoryTheory.Functor C C)) {m n m' n' : M} (f : m ⟶ m') (g : n ⟶ n') (X : C)
Goal: CategoryTheory.CategoryStruct.comp ((F.map g).app ((F.obj m).obj X))
    (CategoryTheory.CategoryStruct.comp ((F.obj n').map ((F.map f).app X)) ((F.μ m' n').app X)) =
  CategoryTheory.CategoryStruct.comp ((F.μ m n).app X) ((F.map (CategoryTheory.MonoidalCategory.tensorHom f g)).app X)