ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctor_toLaxMonoidalFunctor_μ
@[simps]
def endMonoidalStarFunctor : MonoidalFunctor (EndMonoidal (MonoidalSingleObj.star C)) C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C] (X Y : CategoryTheory.EndMonoidal (CategoryTheory.MonoidalSingleObj.star C))
Goal: (CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctor C).μ X Y =
  CategoryTheory.CategoryStruct.id
    (CategoryTheory.MonoidalCategory.tensorObj
      ({ obj := fun X => X, map := fun {X Y} f => f, map_id := ⋯, map_comp := ⋯ }.obj X)
      ({ obj := fun X => X, map := fun {X Y} f => f, map_id := ⋯, map_comp := ⋯ }.obj Y))
-- CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctor_toLaxMonoidalFunctor_ε
@[simps]
def endMonoidalStarFunctor : MonoidalFunctor (EndMonoidal (MonoidalSingleObj.star C)) C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C]
Goal: (CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctor C).ε = CategoryTheory.CategoryStruct.id (𝟙_ C)
-- CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctor_toLaxMonoidalFunctor_toFunctor_obj
@[simps]
def endMonoidalStarFunctor : MonoidalFunctor (EndMonoidal (MonoidalSingleObj.star C)) C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C] (X : CategoryTheory.EndMonoidal (CategoryTheory.MonoidalSingleObj.star C))
Goal: (CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctor C).obj X = X
-- CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctor_toLaxMonoidalFunctor_toFunctor_map
@[simps]
def endMonoidalStarFunctor : MonoidalFunctor (EndMonoidal (MonoidalSingleObj.star C)) C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C]
Goal: ∀ {X Y : CategoryTheory.EndMonoidal (CategoryTheory.MonoidalSingleObj.star C)} (f : X ⟶ Y),
  (CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctor C).map f = f
-- CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctorEquivalence_counitIso
@[simps functor inverse_obj inverse_map unitIso counitIso]
noncomputable def endMonoidalStarFunctorEquivalence :
    EndMonoidal (MonoidalSingleObj.star C) ≌ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C]
Goal: (CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctorEquivalence C).counitIso =
  CategoryTheory.Iso.refl
    ({ obj := fun X => X, map := fun {X Y} f => f, map_id := ⋯, map_comp := ⋯ }.comp
      (CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctor C).toFunctor)
-- CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctorEquivalence_inverse_obj
@[simps functor inverse_obj inverse_map unitIso counitIso]
noncomputable def endMonoidalStarFunctorEquivalence :
    EndMonoidal (MonoidalSingleObj.star C) ≌ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctorEquivalence C).inverse.obj X = X
-- CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctorEquivalence_unitIso
@[simps functor inverse_obj inverse_map unitIso counitIso]
noncomputable def endMonoidalStarFunctorEquivalence :
    EndMonoidal (MonoidalSingleObj.star C) ≌ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C]
Goal: (CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctorEquivalence C).unitIso =
  CategoryTheory.Iso.refl
    (CategoryTheory.Functor.id (CategoryTheory.EndMonoidal (CategoryTheory.MonoidalSingleObj.star C)))
-- CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctorEquivalence_functor
@[simps functor inverse_obj inverse_map unitIso counitIso]
noncomputable def endMonoidalStarFunctorEquivalence :
    EndMonoidal (MonoidalSingleObj.star C) ≌ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C]
Goal: (CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctorEquivalence C).functor =
  (CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctor C).toFunctor
-- CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctorEquivalence_inverse_map
@[simps functor inverse_obj inverse_map unitIso counitIso]
noncomputable def endMonoidalStarFunctorEquivalence :
    EndMonoidal (MonoidalSingleObj.star C) ≌ C 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.MonoidalCategory C]
Goal: ∀ {X Y : C} (f : X ⟶ Y), (CategoryTheory.MonoidalSingleObj.endMonoidalStarFunctorEquivalence C).inverse.map f = f
-- CategoryTheory.StrongOplaxNatTrans.naturality_id
structure StrongOplaxNatTrans (F G : OplaxFunctor B C) where
  app (a : B) : F.obj a ⟶ G.obj a
  naturality {a b : B} (f : a ⟶ b) : F.map f ≫ app b ≅ app a ≫ G.map f
  naturality_naturality :
    ∀ {a b : B} {f g : a ⟶ b} (η : f ⟶ g),
      F.map₂ η ▷ app b ≫ (naturality g).hom = (naturality f).hom ≫ app a ◁ G.map₂ η := by
    aesop_cat
  naturality_id :
    ∀ a : B,
      (naturality (𝟙 a)).hom ≫ app a ◁ G.mapId a =
        F.mapId a ▷ app a ≫ (λ_ (app a)).hom ≫ (ρ_ (app a)).inv := by
    aesop_cat
  naturality_comp :
    ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c),
      (naturality (f ≫ g)).hom ≫ app a ◁ G.mapComp f g =
        F.mapComp f g ▷ app c ≫ (α_ _ _ _).hom ≫ F.map f ◁ (naturality g).hom ≫
        (α_ _ _ _).inv ≫ (naturality f).hom ▷ G.map g ≫ (α_ _ _ _).hom := by
    aesop_cat
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (self : CategoryTheory.StrongOplaxNatTrans F G) (a : B)
Goal: CategoryTheory.CategoryStruct.comp (self.naturality (CategoryTheory.CategoryStruct.id a)).hom
    (CategoryTheory.Bicategory.whiskerLeft (self.app a) (G.mapId a)) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (F.mapId a) (self.app a))
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor (self.app a)).hom
      (CategoryTheory.Bicategory.rightUnitor (self.app a)).inv)
-- CategoryTheory.StrongOplaxNatTrans.naturality_naturality
structure StrongOplaxNatTrans (F G : OplaxFunctor B C) where
  app (a : B) : F.obj a ⟶ G.obj a
  naturality {a b : B} (f : a ⟶ b) : F.map f ≫ app b ≅ app a ≫ G.map f
  naturality_naturality :
    ∀ {a b : B} {f g : a ⟶ b} (η : f ⟶ g),
      F.map₂ η ▷ app b ≫ (naturality g).hom = (naturality f).hom ≫ app a ◁ G.map₂ η := by
    aesop_cat
  naturality_id :
    ∀ a : B,
      (naturality (𝟙 a)).hom ≫ app a ◁ G.mapId a =
        F.mapId a ▷ app a ≫ (λ_ (app a)).hom ≫ (ρ_ (app a)).inv := by
    aesop_cat
  naturality_comp :
    ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c),
      (naturality (f ≫ g)).hom ≫ app a ◁ G.mapComp f g =
        F.mapComp f g ▷ app c ≫ (α_ _ _ _).hom ≫ F.map f ◁ (naturality g).hom ≫
        (α_ _ _ _).inv ≫ (naturality f).hom ▷ G.map g ≫ (α_ _ _ _).hom := by
    aesop_cat
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (self : CategoryTheory.StrongOplaxNatTrans F G) {a b : B} {f g : a ⟶ b} (η : f ⟶ g)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (F.map₂ η) (self.app b))
    (self.naturality g).hom =
  CategoryTheory.CategoryStruct.comp (self.naturality f).hom
    (CategoryTheory.Bicategory.whiskerLeft (self.app a) (G.map₂ η))
-- CategoryTheory.StrongOplaxNatTrans.naturality_comp
structure StrongOplaxNatTrans (F G : OplaxFunctor B C) where
  app (a : B) : F.obj a ⟶ G.obj a
  naturality {a b : B} (f : a ⟶ b) : F.map f ≫ app b ≅ app a ≫ G.map f
  naturality_naturality :
    ∀ {a b : B} {f g : a ⟶ b} (η : f ⟶ g),
      F.map₂ η ▷ app b ≫ (naturality g).hom = (naturality f).hom ≫ app a ◁ G.map₂ η := by
    aesop_cat
  naturality_id :
    ∀ a : B,
      (naturality (𝟙 a)).hom ≫ app a ◁ G.mapId a =
        F.mapId a ▷ app a ≫ (λ_ (app a)).hom ≫ (ρ_ (app a)).inv := by
    aesop_cat
  naturality_comp :
    ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c),
      (naturality (f ≫ g)).hom ≫ app a ◁ G.mapComp f g =
        F.mapComp f g ▷ app c ≫ (α_ _ _ _).hom ≫ F.map f ◁ (naturality g).hom ≫
        (α_ _ _ _).inv ≫ (naturality f).hom ▷ G.map g ≫ (α_ _ _ _).hom := by
    aesop_cat
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (self : CategoryTheory.StrongOplaxNatTrans F G) {a b c : B} (f : a ⟶ b) (g : b ⟶ c)
Goal: CategoryTheory.CategoryStruct.comp (self.naturality (CategoryTheory.CategoryStruct.comp f g)).hom
    (CategoryTheory.Bicategory.whiskerLeft (self.app a) (G.mapComp f g)) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (F.mapComp f g) (self.app c))
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (F.map f) (F.map g) (self.app c)).hom
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (F.map f) (self.naturality g).hom)
        (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (F.map f) (self.app b) (G.map g)).inv
          (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (self.naturality f).hom (G.map g))
            (CategoryTheory.Bicategory.associator (self.app a) (G.map f) (G.map g)).hom))))
-- CategoryTheory.StrongOplaxNatTrans.toOplax_naturality
@[simps]
def toOplax {F G : OplaxFunctor B C} (η : StrongOplaxNatTrans F G) : OplaxNatTrans F G 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.StrongOplaxNatTrans F G)
Goal: ∀ {a b : B} (f : a ⟶ b), η.toOplax.naturality f = (η.naturality f).hom
-- CategoryTheory.StrongOplaxNatTrans.toOplax_app
@[simps]
def toOplax {F G : OplaxFunctor B C} (η : StrongOplaxNatTrans F G) : OplaxNatTrans F G 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.StrongOplaxNatTrans F G) (a : B)
Goal: η.toOplax.app a = η.app a
-- CategoryTheory.StrongOplaxNatTrans.id_naturality_inv
@[simps!]
def id : StrongOplaxNatTrans F F 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F : CategoryTheory.OplaxFunctor B C)
Goal: ∀ {a b : B} (f : a ⟶ b),
  ((CategoryTheory.StrongOplaxNatTrans.id F).naturality f).inv =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor (F.map f)).hom
      (CategoryTheory.Bicategory.rightUnitor (F.map f)).inv
-- CategoryTheory.StrongOplaxNatTrans.id_naturality_hom
@[simps!]
def id : StrongOplaxNatTrans F F 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F : CategoryTheory.OplaxFunctor B C)
Goal: ∀ {a b : B} (f : a ⟶ b),
  ((CategoryTheory.StrongOplaxNatTrans.id F).naturality f).hom =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor (F.map f)).hom
      (CategoryTheory.Bicategory.leftUnitor (F.map f)).inv
-- CategoryTheory.StrongOplaxNatTrans.id_app
@[simps!]
def id : StrongOplaxNatTrans F F 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F : CategoryTheory.OplaxFunctor B C) (a : B)
Goal: (CategoryTheory.StrongOplaxNatTrans.id F).app a = CategoryTheory.CategoryStruct.id (F.obj a)
-- CategoryTheory.StrongOplaxNatTrans.id.toOplax
@[simp]
lemma id.toOplax : (id F).toOplax = OplaxNatTrans.id F 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F : CategoryTheory.OplaxFunctor B C)
Goal: (CategoryTheory.StrongOplaxNatTrans.id F).toOplax = CategoryTheory.OplaxNatTrans.id F
-- CategoryTheory.StrongOplaxNatTrans.whiskerLeft_naturality_naturality
@[reassoc (attr := simp)]
theorem whiskerLeft_naturality_naturality (f : a' ⟶ G.obj a) {g h : a ⟶ b} (β : g ⟶ h) :
    f ◁ G.map₂ β ▷ θ.app b ≫ f ◁ (θ.naturality h).hom =
      f ◁ (θ.naturality g).hom ≫ f ◁ θ.app a ◁ H.map₂ β 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {G H : CategoryTheory.OplaxFunctor B C} (θ : CategoryTheory.StrongOplaxNatTrans G H) {a b : B} {a' : C} (f : a' ⟶ G.obj a) {g h : a ⟶ b} (β : g ⟶ h)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight (G.map₂ β) (θ.app b)))
    (CategoryTheory.Bicategory.whiskerLeft f (θ.naturality h).hom) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft f (θ.naturality g).hom)
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft (θ.app a) (H.map₂ β)))
-- CategoryTheory.StrongOplaxNatTrans.whiskerRight_naturality_naturality
@[reassoc (attr := simp)]
theorem whiskerRight_naturality_naturality {f g : a ⟶ b} (β : f ⟶ g) (h : G.obj b ⟶ a') :
    F.map₂ β ▷ η.app b ▷ h ≫ (η.naturality g).hom ▷ h =
      (η.naturality f).hom ▷ h ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.map₂ β ▷ h ≫ (α_ _ _ _).inv 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.StrongOplaxNatTrans F G) {a b : B} {a' : C} {f g : a ⟶ b} (β : f ⟶ g) (h : G.obj b ⟶ a')
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight (F.map₂ β) (η.app b)) h)
    (CategoryTheory.Bicategory.whiskerRight (η.naturality g).hom h) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (η.naturality f).hom h)
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (η.app a) (G.map f) h).hom
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Bicategory.whiskerLeft (η.app a) (CategoryTheory.Bicategory.whiskerRight (G.map₂ β) h))
        (CategoryTheory.Bicategory.associator (η.app a) (G.map g) h).inv))
-- CategoryTheory.StrongOplaxNatTrans.whiskerLeft_naturality_comp
@[reassoc (attr := simp)]
theorem whiskerLeft_naturality_comp (f : a' ⟶ G.obj a) (g : a ⟶ b) (h : b ⟶ c) :
    f ◁ (θ.naturality (g ≫ h)).hom ≫ f ◁ θ.app a ◁ H.mapComp g h =
      f ◁ G.mapComp g h ▷ θ.app c ≫
        f ◁ (α_ _ _ _).hom ≫
          f ◁ G.map g ◁ (θ.naturality h).hom ≫
            f ◁ (α_ _ _ _).inv ≫ f ◁ (θ.naturality g).hom ▷ H.map h ≫ f ◁ (α_ _ _ _).hom 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {G H : CategoryTheory.OplaxFunctor B C} (θ : CategoryTheory.StrongOplaxNatTrans G H) {a b c : B} {a' : C} (f : a' ⟶ G.obj a) (g : a ⟶ b) (h : b ⟶ c)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (θ.naturality (CategoryTheory.CategoryStruct.comp g h)).hom)
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft (θ.app a) (H.mapComp g h))) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight (G.mapComp g h) (θ.app c)))
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator (G.map g) (G.map h) (θ.app c)).hom)
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft (G.map g) (θ.naturality h).hom))
        (CategoryTheory.CategoryStruct.comp
          (CategoryTheory.Bicategory.whiskerLeft f
            (CategoryTheory.Bicategory.associator (G.map g) (θ.app b) (H.map h)).inv)
          (CategoryTheory.CategoryStruct.comp
            (CategoryTheory.Bicategory.whiskerLeft f
              (CategoryTheory.Bicategory.whiskerRight (θ.naturality g).hom (H.map h)))
            (CategoryTheory.Bicategory.whiskerLeft f
              (CategoryTheory.Bicategory.associator (θ.app a) (H.map g) (H.map h)).hom)))))
-- CategoryTheory.StrongOplaxNatTrans.whiskerRight_naturality_comp
@[reassoc (attr := simp)]
theorem whiskerRight_naturality_comp (f : a ⟶ b) (g : b ⟶ c) (h : G.obj c ⟶ a') :
    (η.naturality (f ≫ g)).hom ▷ h ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.mapComp f g ▷ h =
      F.mapComp f g ▷ η.app c ▷ h ≫
        (α_ _ _ _).hom ▷ h ≫
          (α_ _ _ _).hom ≫
            F.map f ◁ (η.naturality g).hom ▷ h ≫
              (α_ _ _ _).inv ≫
                (α_ _ _ _).inv ▷ h ≫
                 (η.naturality f).hom ▷ G.map g ▷ h ≫ (α_ _ _ _).hom ▷ h ≫ (α_ _ _ _).hom 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.StrongOplaxNatTrans F G) {a b c : B} {a' : C} (f : a ⟶ b) (g : b ⟶ c) (h : G.obj c ⟶ a')
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (η.naturality (CategoryTheory.CategoryStruct.comp f g)).hom h)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator (η.app a) (G.map (CategoryTheory.CategoryStruct.comp f g)) h).hom
      (CategoryTheory.Bicategory.whiskerLeft (η.app a) (CategoryTheory.Bicategory.whiskerRight (G.mapComp f g) h))) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight (F.mapComp f g) (η.app c)) h)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator (F.map f) (F.map g) (η.app c)).hom
        h)
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Bicategory.associator (F.map f) (CategoryTheory.CategoryStruct.comp (F.map g) (η.app c)) h).hom
        (CategoryTheory.CategoryStruct.comp
          (CategoryTheory.Bicategory.whiskerLeft (F.map f)
            (CategoryTheory.Bicategory.whiskerRight (η.naturality g).hom h))
          (CategoryTheory.CategoryStruct.comp
            (CategoryTheory.Bicategory.associator (F.map f) (CategoryTheory.CategoryStruct.comp (η.app b) (G.map g))
                h).inv
            (CategoryTheory.CategoryStruct.comp
              (CategoryTheory.Bicategory.whiskerRight
                (CategoryTheory.Bicategory.associator (F.map f) (η.app b) (G.map g)).inv h)
              (CategoryTheory.CategoryStruct.comp
                (CategoryTheory.Bicategory.whiskerRight
                  (CategoryTheory.Bicategory.whiskerRight (η.naturality f).hom (G.map g)) h)
                (CategoryTheory.CategoryStruct.comp
                  (CategoryTheory.Bicategory.whiskerRight
                    (CategoryTheory.Bicategory.associator (η.app a) (G.map f) (G.map g)).hom h)
                  (CategoryTheory.Bicategory.associator (η.app a)
                      (CategoryTheory.CategoryStruct.comp (G.map f) (G.map g)) h).hom)))))))
-- CategoryTheory.StrongOplaxNatTrans.whiskerLeft_naturality_id
@[reassoc (attr := simp)]
theorem whiskerLeft_naturality_id (f : a' ⟶ G.obj a) :
    f ◁ (θ.naturality (𝟙 a)).hom ≫ f ◁ θ.app a ◁ H.mapId a =
      f ◁ G.mapId a ▷ θ.app a ≫ f ◁ (λ_ (θ.app a)).hom ≫ f ◁ (ρ_ (θ.app a)).inv 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {G H : CategoryTheory.OplaxFunctor B C} (θ : CategoryTheory.StrongOplaxNatTrans G H) {a : B} {a' : C} (f : a' ⟶ G.obj a)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (θ.naturality (CategoryTheory.CategoryStruct.id a)).hom)
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft (θ.app a) (H.mapId a))) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight (G.mapId a) (θ.app a)))
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.leftUnitor (θ.app a)).hom)
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.rightUnitor (θ.app a)).inv))
-- CategoryTheory.StrongOplaxNatTrans.whiskerRight_naturality_id
@[reassoc (attr := simp)]
theorem whiskerRight_naturality_id (f : G.obj a ⟶ a') :
    (η.naturality (𝟙 a)).hom ▷ f ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.mapId a ▷ f =
    F.mapId a ▷ η.app a ▷ f ≫ (λ_ (η.app a)).hom ▷ f ≫ (ρ_ (η.app a)).inv ▷ f ≫
    (α_ _ _ _).hom 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.StrongOplaxNatTrans F G) {a : B} {a' : C} (f : G.obj a ⟶ a')
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (η.naturality (CategoryTheory.CategoryStruct.id a)).hom f)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator (η.app a) (G.map (CategoryTheory.CategoryStruct.id a)) f).hom
      (CategoryTheory.Bicategory.whiskerLeft (η.app a) (CategoryTheory.Bicategory.whiskerRight (G.mapId a) f))) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight (F.mapId a) (η.app a)) f)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.leftUnitor (η.app a)).hom f)
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.rightUnitor (η.app a)).inv f)
        (CategoryTheory.Bicategory.associator (η.app a) (CategoryTheory.CategoryStruct.id (G.obj a)) f).hom))
-- CategoryTheory.StrongOplaxNatTrans.vcomp_naturality_inv
@[simps!]
def vcomp (η : StrongOplaxNatTrans F G) (θ : StrongOplaxNatTrans G H) : StrongOplaxNatTrans F H 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G H : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.StrongOplaxNatTrans F G) (θ : CategoryTheory.StrongOplaxNatTrans G H)
Goal: ∀ {a b : B} (f : a ⟶ b),
  ((η.vcomp θ).naturality f).inv =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (η.app a) (θ.app a) (H.map f)).hom
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (η.app a) (θ.naturality f).inv)
        (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (η.app a) (G.map f) (θ.app b)).inv
          (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (η.naturality f).inv (θ.app b))
            (CategoryTheory.Bicategory.associator (F.map f) (η.app b) (θ.app b)).hom)))
-- CategoryTheory.StrongOplaxNatTrans.vcomp_app
@[simps!]
def vcomp (η : StrongOplaxNatTrans F G) (θ : StrongOplaxNatTrans G H) : StrongOplaxNatTrans F H 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G H : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.StrongOplaxNatTrans F G) (θ : CategoryTheory.StrongOplaxNatTrans G H) (a : B)
Goal: (η.vcomp θ).app a = CategoryTheory.CategoryStruct.comp (η.app a) (θ.app a)
-- CategoryTheory.StrongOplaxNatTrans.vcomp_naturality_hom
@[simps!]
def vcomp (η : StrongOplaxNatTrans F G) (θ : StrongOplaxNatTrans G H) : StrongOplaxNatTrans F H 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G H : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.StrongOplaxNatTrans F G) (θ : CategoryTheory.StrongOplaxNatTrans G H)
Goal: ∀ {a b : B} (f : a ⟶ b),
  ((η.vcomp θ).naturality f).hom =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (F.map f) (η.app b) (θ.app b)).inv
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (η.naturality f).hom (θ.app b))
        (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (η.app a) (G.map f) (θ.app b)).hom
          (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (η.app a) (θ.naturality f).hom)
            (CategoryTheory.Bicategory.associator (η.app a) (θ.app a) (H.map f)).inv)))
-- CategoryTheory.Pseudofunctor.categoryStruct_comp
@[simps id comp]
instance Pseudofunctor.categoryStruct : CategoryStruct (Pseudofunctor B C) 
Arguments: (B : Type u₁) [CategoryTheory.Bicategory B] (C : Type u₂) [CategoryTheory.Bicategory C]
Goal: ∀ {X Y Z : CategoryTheory.Pseudofunctor B C} (η : CategoryTheory.StrongOplaxNatTrans X.toOplax Y.toOplax)
  (θ : CategoryTheory.StrongOplaxNatTrans Y.toOplax Z.toOplax), CategoryTheory.CategoryStruct.comp η θ = η.vcomp θ