File: Mathlib/CategoryTheory/Bicategory/Basic.lean

full_name: CategoryTheory.Bicategory.comp_whiskerLeft
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.comp f g) η =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).hom
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft g η))
      (CategoryTheory.Bicategory.associator f g h').inv)

full_name: CategoryTheory.Bicategory.whiskerLeft_comp
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.CategoryStruct.comp η θ) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft f η)
    (CategoryTheory.Bicategory.whiskerLeft f θ)

full_name: CategoryTheory.Bicategory.id_whiskerLeft
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.id a) η =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor f).hom
    (CategoryTheory.CategoryStruct.comp η (CategoryTheory.Bicategory.leftUnitor g).inv)

full_name: CategoryTheory.Bicategory.whisker_assoc
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerLeft f η) h =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).hom
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight η h))
      (CategoryTheory.Bicategory.associator f g' h).inv)

full_name: CategoryTheory.Bicategory.rightUnitor
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.id b) ≅ f

full_name: CategoryTheory.Bicategory.id_whiskerRight
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.Bicategory.whiskerRight (CategoryTheory.CategoryStruct.id f) g =
  CategoryTheory.CategoryStruct.id (CategoryTheory.CategoryStruct.comp f g)

full_name: CategoryTheory.Bicategory.homCategory
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.Category.{w, v} (a ⟶ b)

full_name: CategoryTheory.Bicategory.whiskerRight_comp
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.comp g h) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).inv
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight η g) h)
      (CategoryTheory.Bicategory.associator f' g h).hom)

full_name: CategoryTheory.Bicategory.whiskerRight_id
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.id b) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor f).hom
    (CategoryTheory.CategoryStruct.comp η (CategoryTheory.Bicategory.rightUnitor g).inv)

full_name: CategoryTheory.Bicategory
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: Type (max (max u (v + 1)) (w + 1))

full_name: CategoryTheory.Bicategory.leftUnitor
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id a) f ≅ f

full_name: CategoryTheory.Bicategory.pentagon
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).hom i)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).hom
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).hom)) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).hom
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).hom

full_name: CategoryTheory.Bicategory.whiskerLeft
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.CategoryStruct.comp f g ⟶ CategoryTheory.CategoryStruct.comp f h

full_name: CategoryTheory.Bicategory.whiskerRight
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.CategoryStruct.comp f h ⟶ CategoryTheory.CategoryStruct.comp g h

full_name: CategoryTheory.Bicategory.associator
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp f g) h ≅
  CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp g h)

full_name: CategoryTheory.Bicategory.whisker_exchange
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft f θ)
    (CategoryTheory.Bicategory.whiskerRight η i) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight η h)
    (CategoryTheory.Bicategory.whiskerLeft g θ)

full_name: CategoryTheory.Bicategory.triangle
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.id b) g).hom
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.leftUnitor g).hom) =
  CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.rightUnitor f).hom g

full_name: CategoryTheory.Bicategory.whiskerLeft_id
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.CategoryStruct.id g) =
  CategoryTheory.CategoryStruct.id (CategoryTheory.CategoryStruct.comp f g)

full_name: CategoryTheory.Bicategory.comp_whiskerRight
code: /-- In a bicategory, we can compose the 1-morphisms `f : a ⟶ b` and `g : b ⟶ c` to obtain
a 1-morphism `f ≫ g : a ⟶ c`. This composition does not need to be strictly associative,
but there is a specified associator, `α_ f g h : (f ≫ g) ≫ h ≅ f ≫ (g ≫ h)`.
There is an identity 1-morphism `𝟙 a : a ⟶ a`, with specified left and right unitor
isomorphisms `λ_ f : 𝟙 a ≫ f ≅ f` and `ρ_ f : f ≫ 𝟙 a ≅ f`.
These associators and unitors satisfy the pentagon and triangle equations.

See https://ncatlab.org/nlab/show/bicategory.
-/
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
  -- category structure on the collection of 1-morphisms:
  homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
  -- left whiskering:
  whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
  -- right whiskering:
  whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
  -- associator:
  associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
  -- left unitor:
  leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
  -- right unitor:
  rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
  -- axioms for left whiskering:
  whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
    aesop_cat
  whiskerLeft_comp :
    ∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
      whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
    aesop_cat
  id_whiskerLeft :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
    aesop_cat
  comp_whiskerLeft :
    ∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
      whiskerLeft (f ≫ g) η =
        (associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
    aesop_cat
  -- axioms for right whiskering:
  id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
    aesop_cat
  comp_whiskerRight :
    ∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
      whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
    aesop_cat
  whiskerRight_id :
    ∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
      whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
    aesop_cat
  whiskerRight_comp :
    ∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
      whiskerRight η (g ≫ h) =
        (associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
    aesop_cat
  -- associativity of whiskerings:
  whisker_assoc :
    ∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
      whiskerRight (whiskerLeft f η) h =
        (associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
    aesop_cat
  -- exchange law of left and right whiskerings:
  whisker_exchange :
    ∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
      whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
    aesop_cat
  -- pentagon identity:
  pentagon :
    ∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
      whiskerRight (associator f g h).hom i ≫
          (associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
        (associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
    aesop_cat
  -- triangle identity:
  triangle :
    ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
      (associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
      = whiskerRight (rightUnitor f).hom g := by
    aesop_cat
goal: CategoryTheory.Bicategory.whiskerRight (CategoryTheory.CategoryStruct.comp η θ) i =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight η i)
    (CategoryTheory.Bicategory.whiskerRight θ i)

full_name: CategoryTheory.Bicategory._aux_Mathlib_CategoryTheory_Bicategory_Basic___unexpand_CategoryTheory_Bicategory_whiskerLeft_1
code: scoped infixr:81 " ◁ " => Bicategory.whiskerLeft
goal: Lean.PrettyPrinter.Unexpander

full_name: CategoryTheory.Bicategory.«_aux_Mathlib_CategoryTheory_Bicategory_Basic___macroRules_CategoryTheory_Bicategory_term_◁__1»
code: scoped infixr:81 " ◁ " => Bicategory.whiskerLeft
goal: Lean.Macro

full_name: CategoryTheory.Bicategory.«_aux_Mathlib_CategoryTheory_Bicategory_Basic___macroRules_CategoryTheory_Bicategory_term_▷__1»
code: scoped infixl:81 " ▷ " => Bicategory.whiskerRight
goal: Lean.Macro

full_name: CategoryTheory.Bicategory._aux_Mathlib_CategoryTheory_Bicategory_Basic___unexpand_CategoryTheory_Bicategory_whiskerRight_1
code: scoped infixl:81 " ▷ " => Bicategory.whiskerRight
goal: Lean.PrettyPrinter.Unexpander

full_name: CategoryTheory.Bicategory._aux_Mathlib_CategoryTheory_Bicategory_Basic___unexpand_CategoryTheory_Bicategory_associator_1
code: scoped notation "α_" => Bicategory.associator
goal: Lean.PrettyPrinter.Unexpander

full_name: CategoryTheory.Bicategory._aux_Mathlib_CategoryTheory_Bicategory_Basic___macroRules_CategoryTheory_Bicategory_termα__1
code: scoped notation "α_" => Bicategory.associator
goal: Lean.Macro

full_name: CategoryTheory.Bicategory._aux_Mathlib_CategoryTheory_Bicategory_Basic___unexpand_CategoryTheory_Bicategory_leftUnitor_1
code: scoped notation "λ_" => Bicategory.leftUnitor
goal: Lean.PrettyPrinter.Unexpander

full_name: CategoryTheory.Bicategory.«_aux_Mathlib_CategoryTheory_Bicategory_Basic___macroRules_CategoryTheory_Bicategory_termλ__1»
code: scoped notation "λ_" => Bicategory.leftUnitor
goal: Lean.Macro

full_name: CategoryTheory.Bicategory._aux_Mathlib_CategoryTheory_Bicategory_Basic___macroRules_CategoryTheory_Bicategory_termρ__1
code: scoped notation "ρ_" => Bicategory.rightUnitor
goal: Lean.Macro

full_name: CategoryTheory.Bicategory._aux_Mathlib_CategoryTheory_Bicategory_Basic___unexpand_CategoryTheory_Bicategory_rightUnitor_1
code: scoped notation "ρ_" => Bicategory.rightUnitor
goal: Lean.PrettyPrinter.Unexpander

full_name: CategoryTheory.Bicategory.whiskerLeft_hom_inv
code: @[reassoc (attr := simp)]
theorem whiskerLeft_hom_inv (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) :
    f ◁ η.hom ≫ f ◁ η.inv = 𝟙 (f ≫ g) 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft f η.hom)
    (CategoryTheory.Bicategory.whiskerLeft f η.inv) =
  CategoryTheory.CategoryStruct.id (CategoryTheory.CategoryStruct.comp f g)

full_name: CategoryTheory.Bicategory.hom_inv_whiskerRight
code: @[reassoc (attr := simp)]
theorem hom_inv_whiskerRight {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) :
    η.hom ▷ h ≫ η.inv ▷ h = 𝟙 (f ≫ h) 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight η.hom h)
    (CategoryTheory.Bicategory.whiskerRight η.inv h) =
  CategoryTheory.CategoryStruct.id (CategoryTheory.CategoryStruct.comp f h)

full_name: CategoryTheory.Bicategory.whiskerLeft_inv_hom
code: @[reassoc (attr := simp)]
theorem whiskerLeft_inv_hom (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) :
    f ◁ η.inv ≫ f ◁ η.hom = 𝟙 (f ≫ h) 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft f η.inv)
    (CategoryTheory.Bicategory.whiskerLeft f η.hom) =
  CategoryTheory.CategoryStruct.id (CategoryTheory.CategoryStruct.comp f h)

full_name: CategoryTheory.Bicategory.inv_hom_whiskerRight
code: @[reassoc (attr := simp)]
theorem inv_hom_whiskerRight {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) :
    η.inv ▷ h ≫ η.hom ▷ h = 𝟙 (g ≫ h) 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight η.inv h)
    (CategoryTheory.Bicategory.whiskerRight η.hom h) =
  CategoryTheory.CategoryStruct.id (CategoryTheory.CategoryStruct.comp g h)

full_name: CategoryTheory.Bicategory.whiskerLeftIso
code: /-- The left whiskering of a 2-isomorphism is a 2-isomorphism. -/
@[simps]
def whiskerLeftIso (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) : f ≫ g ≅ f ≫ h 
goal: CategoryTheory.CategoryStruct.comp f g ≅ CategoryTheory.CategoryStruct.comp f h

full_name: CategoryTheory.Bicategory.whiskerLeftIso_inv
code: /-- The left whiskering of a 2-isomorphism is a 2-isomorphism. -/
@[simps]
def whiskerLeftIso (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) : f ≫ g ≅ f ≫ h 
goal: (CategoryTheory.Bicategory.whiskerLeftIso f η).inv = CategoryTheory.Bicategory.whiskerLeft f η.inv

full_name: CategoryTheory.Bicategory.whiskerLeftIso_hom
code: /-- The left whiskering of a 2-isomorphism is a 2-isomorphism. -/
@[simps]
def whiskerLeftIso (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) : f ≫ g ≅ f ≫ h 
goal: (CategoryTheory.Bicategory.whiskerLeftIso f η).hom = CategoryTheory.Bicategory.whiskerLeft f η.hom

full_name: CategoryTheory.Bicategory.whiskerLeft_isIso
code: instance whiskerLeft_isIso (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) [IsIso η] : IsIso (f ◁ η) 
goal: CategoryTheory.IsIso (CategoryTheory.Bicategory.whiskerLeft f η)

full_name: CategoryTheory.Bicategory.inv_whiskerLeft
code: @[simp]
theorem inv_whiskerLeft (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) [IsIso η] :
    inv (f ◁ η) = f ◁ inv η 
goal: CategoryTheory.inv (CategoryTheory.Bicategory.whiskerLeft f η) =
  CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.inv η)

full_name: CategoryTheory.Bicategory.whiskerRightIso_hom
code: /-- The right whiskering of a 2-isomorphism is a 2-isomorphism. -/
@[simps!]
def whiskerRightIso {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) : f ≫ h ≅ g ≫ h 
goal: (CategoryTheory.Bicategory.whiskerRightIso η h).hom = CategoryTheory.Bicategory.whiskerRight η.hom h

full_name: CategoryTheory.Bicategory.whiskerRightIso
code: /-- The right whiskering of a 2-isomorphism is a 2-isomorphism. -/
@[simps!]
def whiskerRightIso {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) : f ≫ h ≅ g ≫ h 
goal: CategoryTheory.CategoryStruct.comp f h ≅ CategoryTheory.CategoryStruct.comp g h

full_name: CategoryTheory.Bicategory.whiskerRightIso_inv
code: /-- The right whiskering of a 2-isomorphism is a 2-isomorphism. -/
@[simps!]
def whiskerRightIso {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) : f ≫ h ≅ g ≫ h 
goal: (CategoryTheory.Bicategory.whiskerRightIso η h).inv = CategoryTheory.Bicategory.whiskerRight η.inv h

full_name: CategoryTheory.Bicategory.whiskerRight_isIso
code: instance whiskerRight_isIso {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) [IsIso η] : IsIso (η ▷ h) 
goal: CategoryTheory.IsIso (CategoryTheory.Bicategory.whiskerRight η h)

full_name: CategoryTheory.Bicategory.inv_whiskerRight
code: @[simp]
theorem inv_whiskerRight {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) [IsIso η] :
    inv (η ▷ h) = inv η ▷ h 
goal: CategoryTheory.inv (CategoryTheory.Bicategory.whiskerRight η h) =
  CategoryTheory.Bicategory.whiskerRight (CategoryTheory.inv η) h

full_name: CategoryTheory.Bicategory.pentagon_inv
code: @[reassoc (attr := simp)]
theorem pentagon_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i =
      (α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).inv)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).inv
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).inv i)) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).inv
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).inv

full_name: CategoryTheory.Bicategory.pentagon_inv_inv_hom_hom_inv
code: @[reassoc (attr := simp)]
theorem pentagon_inv_inv_hom_hom_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom =
    f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).inv
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).inv i)
      (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).hom) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).hom)
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).inv

full_name: CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_inv
code: @[reassoc (attr := simp)]
theorem pentagon_inv_hom_hom_hom_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i ≫ (α_ f (g ≫ h) i).hom =
      (α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).inv
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).hom i)
      (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).hom) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).hom
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).inv)

full_name: CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_inv
code: @[reassoc (attr := simp)]
theorem pentagon_hom_inv_inv_inv_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv =
      (α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).hom)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).inv
      (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).inv) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).inv
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).inv i)

full_name: CategoryTheory.Bicategory.pentagon_hom_hom_inv_hom_hom
code: @[reassoc (attr := simp)]
theorem pentagon_hom_hom_inv_hom_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ (f ≫ g) h i).hom ≫ (α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv =
      (α_ f g h).hom ▷ i ≫ (α_ f (g ≫ h) i).hom 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).hom
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).hom
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).inv)) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).hom i)
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).hom

full_name: CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_hom
code: @[reassoc (attr := simp)]
theorem pentagon_hom_inv_inv_inv_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv =
    (α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).hom
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).inv)
      (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).inv) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).inv
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).hom i)

full_name: CategoryTheory.Bicategory.pentagon_hom_hom_inv_inv_hom
code: @[reassoc (attr := simp)]
theorem pentagon_hom_hom_inv_inv_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ f (g ≫ h) i).hom ≫ f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv =
      (α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).hom
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).hom)
      (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).inv) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).inv i)
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).hom

full_name: CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_hom
code: @[reassoc (attr := simp)]
theorem pentagon_inv_hom_hom_hom_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom ≫ (α_ f g (h ≫ i)).hom =
      (α_ f (g ≫ h) i).hom ≫ f ◁ (α_ g h i).hom 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).inv i)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).hom
      (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).hom) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).hom
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).hom)

full_name: CategoryTheory.Bicategory.pentagon_inv_inv_hom_inv_inv
code: @[reassoc (attr := simp)]
theorem pentagon_inv_inv_hom_inv_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i =
      f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).inv
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).inv
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).hom i)) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).inv)
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).inv

full_name: CategoryTheory.Bicategory.triangle_assoc_comp_left
code: theorem triangle_assoc_comp_left (f : a ⟶ b) (g : b ⟶ c) :
    (α_ f (𝟙 b) g).hom ≫ f ◁ (λ_ g).hom = (ρ_ f).hom ▷ g 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.id b) g).hom
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.leftUnitor g).hom) =
  CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.rightUnitor f).hom g

full_name: CategoryTheory.Bicategory.triangle_assoc_comp_right
code: @[reassoc (attr := simp)]
theorem triangle_assoc_comp_right (f : a ⟶ b) (g : b ⟶ c) :
    (α_ f (𝟙 b) g).inv ≫ (ρ_ f).hom ▷ g = f ◁ (λ_ g).hom 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.id b) g).inv
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.rightUnitor f).hom g) =
  CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.leftUnitor g).hom

full_name: CategoryTheory.Bicategory.triangle_assoc_comp_right_inv
code: @[reassoc (attr := simp)]
theorem triangle_assoc_comp_right_inv (f : a ⟶ b) (g : b ⟶ c) :
    (ρ_ f).inv ▷ g ≫ (α_ f (𝟙 b) g).hom = f ◁ (λ_ g).inv 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.rightUnitor f).inv g)
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.id b) g).hom =
  CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.leftUnitor g).inv

full_name: CategoryTheory.Bicategory.triangle_assoc_comp_left_inv
code: @[reassoc (attr := simp)]
theorem triangle_assoc_comp_left_inv (f : a ⟶ b) (g : b ⟶ c) :
    f ◁ (λ_ g).inv ≫ (α_ f (𝟙 b) g).inv = (ρ_ f).inv ▷ g 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.leftUnitor g).inv)
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.id b) g).inv =
  CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.rightUnitor f).inv g

full_name: CategoryTheory.Bicategory.associator_naturality_left
code: @[reassoc]
theorem associator_naturality_left {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
    η ▷ g ▷ h ≫ (α_ f' g h).hom = (α_ f g h).hom ≫ η ▷ (g ≫ h) 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight η g) h)
    (CategoryTheory.Bicategory.associator f' g h).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).hom
    (CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.comp g h))

full_name: CategoryTheory.Bicategory.associator_inv_naturality_left
code: @[reassoc]
theorem associator_inv_naturality_left {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
    η ▷ (g ≫ h) ≫ (α_ f' g h).inv = (α_ f g h).inv ≫ η ▷ g ▷ h 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.comp g h))
    (CategoryTheory.Bicategory.associator f' g h).inv =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).inv
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight η g) h)

full_name: CategoryTheory.Bicategory.whiskerRight_comp_symm
code: @[reassoc]
theorem whiskerRight_comp_symm {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
    η ▷ g ▷ h = (α_ f g h).hom ≫ η ▷ (g ≫ h) ≫ (α_ f' g h).inv 
goal: CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight η g) h =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).hom
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.comp g h))
      (CategoryTheory.Bicategory.associator f' g h).inv)

full_name: CategoryTheory.Bicategory.associator_naturality_middle
code: @[reassoc]
theorem associator_naturality_middle (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
    (f ◁ η) ▷ h ≫ (α_ f g' h).hom = (α_ f g h).hom ≫ f ◁ η ▷ h 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerLeft f η) h)
    (CategoryTheory.Bicategory.associator f g' h).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).hom
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight η h))

full_name: CategoryTheory.Bicategory.associator_inv_naturality_middle
code: @[reassoc]
theorem associator_inv_naturality_middle (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
    f ◁ η ▷ h ≫ (α_ f g' h).inv = (α_ f g h).inv ≫ (f ◁ η) ▷ h 
goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight η h))
    (CategoryTheory.Bicategory.associator f g' h).inv =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).inv
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerLeft f η) h)

full_name: CategoryTheory.Bicategory.whisker_assoc_symm
code: @[reassoc]
theorem whisker_assoc_symm (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
    f ◁ η ▷ h = (α_ f g h).inv ≫ (f ◁ η) ▷ h ≫ (α_ f g' h).hom 
goal: CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight η h) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).inv
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerLeft f η) h)
      (CategoryTheory.Bicategory.associator f g' h).hom)

full_name: CategoryTheory.Bicategory.associator_naturality_right
code: @[reassoc]
theorem associator_naturality_right (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h') :
    (f ≫ g) ◁ η ≫ (α_ f g h').hom = (α_ f g h).hom ≫ f ◁ g ◁ η 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.comp f g) η)
    (CategoryTheory.Bicategory.associator f g h').hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).hom
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft g η))

full_name: CategoryTheory.Bicategory.associator_inv_naturality_right
code: @[reassoc]
theorem associator_inv_naturality_right (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h') :
    f ◁ g ◁ η ≫ (α_ f g h').inv = (α_ f g h).inv ≫ (f ≫ g) ◁ η 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft g η))
    (CategoryTheory.Bicategory.associator f g h').inv =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).inv
    (CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.comp f g) η)

full_name: CategoryTheory.Bicategory.comp_whiskerLeft_symm
code: @[reassoc]
theorem comp_whiskerLeft_symm (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h') :
    f ◁ g ◁ η = (α_ f g h).inv ≫ (f ≫ g) ◁ η ≫ (α_ f g h').hom 
goal: CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft g η) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).inv
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.comp f g) η)
      (CategoryTheory.Bicategory.associator f g h').hom)

full_name: CategoryTheory.Bicategory.leftUnitor_naturality
code: @[reassoc]
theorem leftUnitor_naturality {f g : a ⟶ b} (η : f ⟶ g) :
    𝟙 a ◁ η ≫ (λ_ g).hom = (λ_ f).hom ≫ η 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.id a) η)
    (CategoryTheory.Bicategory.leftUnitor g).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor f).hom η

full_name: CategoryTheory.Bicategory.leftUnitor_inv_naturality
code: @[reassoc]
theorem leftUnitor_inv_naturality {f g : a ⟶ b} (η : f ⟶ g) :
    η ≫ (λ_ g).inv = (λ_ f).inv ≫ 𝟙 a ◁ η 
goal: CategoryTheory.CategoryStruct.comp η (CategoryTheory.Bicategory.leftUnitor g).inv =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor f).inv
    (CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.id a) η)

full_name: CategoryTheory.Bicategory.id_whiskerLeft_symm
code: theorem id_whiskerLeft_symm {f g : a ⟶ b} (η : f ⟶ g) : η = (λ_ f).inv ≫ 𝟙 a ◁ η ≫ (λ_ g).hom 
goal: η =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor f).inv
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.id a) η)
      (CategoryTheory.Bicategory.leftUnitor g).hom)

full_name: CategoryTheory.Bicategory.rightUnitor_naturality
code: @[reassoc]
theorem rightUnitor_naturality {f g : a ⟶ b} (η : f ⟶ g) :
    η ▷ 𝟙 b ≫ (ρ_ g).hom = (ρ_ f).hom ≫ η 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.id b))
    (CategoryTheory.Bicategory.rightUnitor g).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor f).hom η

full_name: CategoryTheory.Bicategory.rightUnitor_inv_naturality
code: @[reassoc]
theorem rightUnitor_inv_naturality {f g : a ⟶ b} (η : f ⟶ g) :
    η ≫ (ρ_ g).inv = (ρ_ f).inv ≫ η ▷ 𝟙 b 
goal: CategoryTheory.CategoryStruct.comp η (CategoryTheory.Bicategory.rightUnitor g).inv =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor f).inv
    (CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.id b))

full_name: CategoryTheory.Bicategory.whiskerRight_id_symm
code: theorem whiskerRight_id_symm {f g : a ⟶ b} (η : f ⟶ g) : η = (ρ_ f).inv ≫ η ▷ 𝟙 b ≫ (ρ_ g).hom 
goal: η =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor f).inv
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.id b))
      (CategoryTheory.Bicategory.rightUnitor g).hom)

full_name: CategoryTheory.Bicategory.whiskerLeft_iff
code: theorem whiskerLeft_iff {f g : a ⟶ b} (η θ : f ⟶ g) : 𝟙 a ◁ η = 𝟙 a ◁ θ ↔ η = θ 
goal: CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.id a) η =
    CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.id a) θ ↔
  η = θ

full_name: CategoryTheory.Bicategory.whiskerRight_iff
code: theorem whiskerRight_iff {f g : a ⟶ b} (η θ : f ⟶ g) : η ▷ 𝟙 b = θ ▷ 𝟙 b ↔ η = θ 
goal: CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.id b) =
    CategoryTheory.Bicategory.whiskerRight θ (CategoryTheory.CategoryStruct.id b) ↔
  η = θ

full_name: CategoryTheory.Bicategory.leftUnitor_whiskerRight
code: /-- We state it as a simp lemma, which is regarded as an involved version of
`id_whiskerRight f g : 𝟙 f ▷ g = 𝟙 (f ≫ g)`.
-/
@[reassoc, simp]
theorem leftUnitor_whiskerRight (f : a ⟶ b) (g : b ⟶ c) :
    (λ_ f).hom ▷ g = (α_ (𝟙 a) f g).hom ≫ (λ_ (f ≫ g)).hom 
goal: CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.leftUnitor f).hom g =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.id a) f g).hom
    (CategoryTheory.Bicategory.leftUnitor (CategoryTheory.CategoryStruct.comp f g)).hom

full_name: CategoryTheory.Bicategory.leftUnitor_inv_whiskerRight
code: @[reassoc, simp]
theorem leftUnitor_inv_whiskerRight (f : a ⟶ b) (g : b ⟶ c) :
    (λ_ f).inv ▷ g = (λ_ (f ≫ g)).inv ≫ (α_ (𝟙 a) f g).inv 
goal: CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.leftUnitor f).inv g =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor (CategoryTheory.CategoryStruct.comp f g)).inv
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.id a) f g).inv

full_name: CategoryTheory.Bicategory.whiskerLeft_rightUnitor
code: @[reassoc, simp]
theorem whiskerLeft_rightUnitor (f : a ⟶ b) (g : b ⟶ c) :
    f ◁ (ρ_ g).hom = (α_ f g (𝟙 c)).inv ≫ (ρ_ (f ≫ g)).hom 
goal: CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.rightUnitor g).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.id c)).inv
    (CategoryTheory.Bicategory.rightUnitor (CategoryTheory.CategoryStruct.comp f g)).hom

full_name: CategoryTheory.Bicategory.whiskerLeft_rightUnitor_inv
code: @[reassoc, simp]
theorem whiskerLeft_rightUnitor_inv (f : a ⟶ b) (g : b ⟶ c) :
    f ◁ (ρ_ g).inv = (ρ_ (f ≫ g)).inv ≫ (α_ f g (𝟙 c)).hom 
goal: CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.rightUnitor g).inv =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.rightUnitor (CategoryTheory.CategoryStruct.comp f g)).inv
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.id c)).hom

full_name: CategoryTheory.Bicategory.leftUnitor_comp
code: @[reassoc]
theorem leftUnitor_comp (f : a ⟶ b) (g : b ⟶ c) :
    (λ_ (f ≫ g)).hom = (α_ (𝟙 a) f g).inv ≫ (λ_ f).hom ▷ g 
goal: (CategoryTheory.Bicategory.leftUnitor (CategoryTheory.CategoryStruct.comp f g)).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.id a) f g).inv
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.leftUnitor f).hom g)

full_name: CategoryTheory.Bicategory.leftUnitor_comp_inv
code: @[reassoc]
theorem leftUnitor_comp_inv (f : a ⟶ b) (g : b ⟶ c) :
    (λ_ (f ≫ g)).inv = (λ_ f).inv ▷ g ≫ (α_ (𝟙 a) f g).hom 
goal: (CategoryTheory.Bicategory.leftUnitor (CategoryTheory.CategoryStruct.comp f g)).inv =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.leftUnitor f).inv g)
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.id a) f g).hom

full_name: CategoryTheory.Bicategory.rightUnitor_comp
code: @[reassoc]
theorem rightUnitor_comp (f : a ⟶ b) (g : b ⟶ c) :
    (ρ_ (f ≫ g)).hom = (α_ f g (𝟙 c)).hom ≫ f ◁ (ρ_ g).hom 
goal: (CategoryTheory.Bicategory.rightUnitor (CategoryTheory.CategoryStruct.comp f g)).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.id c)).hom
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.rightUnitor g).hom)

full_name: CategoryTheory.Bicategory.rightUnitor_comp_inv
code: @[reassoc]
theorem rightUnitor_comp_inv (f : a ⟶ b) (g : b ⟶ c) :
    (ρ_ (f ≫ g)).inv = f ◁ (ρ_ g).inv ≫ (α_ f g (𝟙 c)).inv 
goal: (CategoryTheory.Bicategory.rightUnitor (CategoryTheory.CategoryStruct.comp f g)).inv =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.rightUnitor g).inv)
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.id c)).inv

full_name: CategoryTheory.Bicategory.unitors_equal
code: @[simp]
theorem unitors_equal : (λ_ (𝟙 a)).hom = (ρ_ (𝟙 a)).hom 
goal: (CategoryTheory.Bicategory.leftUnitor (CategoryTheory.CategoryStruct.id a)).hom =
  (CategoryTheory.Bicategory.rightUnitor (CategoryTheory.CategoryStruct.id a)).hom

full_name: CategoryTheory.Bicategory.unitors_inv_equal
code: @[simp]
theorem unitors_inv_equal : (λ_ (𝟙 a)).inv = (ρ_ (𝟙 a)).inv 
goal: (CategoryTheory.Bicategory.leftUnitor (CategoryTheory.CategoryStruct.id a)).inv =
  (CategoryTheory.Bicategory.rightUnitor (CategoryTheory.CategoryStruct.id a)).inv

full_name: CategoryTheory.Bicategory.precomp_obj
code: /-- Precomposition of a 1-morphism as a functor. -/
@[simps]
def precomp (c : B) (f : a ⟶ b) : (b ⟶ c) ⥤ (a ⟶ c) 
goal: ∀ (x : b ⟶ c), (CategoryTheory.Bicategory.precomp c f).obj x = CategoryTheory.CategoryStruct.comp f x

full_name: CategoryTheory.Bicategory.precomp
code: /-- Precomposition of a 1-morphism as a functor. -/
@[simps]
def precomp (c : B) (f : a ⟶ b) : (b ⟶ c) ⥤ (a ⟶ c) 
goal: CategoryTheory.Functor (b ⟶ c) (a ⟶ c)

full_name: CategoryTheory.Bicategory.precomp_map
code: /-- Precomposition of a 1-morphism as a functor. -/
@[simps]
def precomp (c : B) (f : a ⟶ b) : (b ⟶ c) ⥤ (a ⟶ c) 
goal: ∀ {X Y : b ⟶ c} (x : X ⟶ Y), (CategoryTheory.Bicategory.precomp c f).map x = CategoryTheory.Bicategory.whiskerLeft f x

full_name: CategoryTheory.Bicategory.precomposing_map_app
code: /-- Precomposition of a 1-morphism as a functor from the category of 1-morphisms `a ⟶ b` into the
category of functors `(b ⟶ c) ⥤ (a ⟶ c)`. -/
@[simps]
def precomposing (a b c : B) : (a ⟶ b) ⥤ (b ⟶ c) ⥤ (a ⟶ c) 
goal: ∀ {X Y : a ⟶ b} (η : X ⟶ Y) (x : b ⟶ c),
  ((CategoryTheory.Bicategory.precomposing a b c).map η).app x = CategoryTheory.Bicategory.whiskerRight η x

full_name: CategoryTheory.Bicategory.precomposing
code: /-- Precomposition of a 1-morphism as a functor from the category of 1-morphisms `a ⟶ b` into the
category of functors `(b ⟶ c) ⥤ (a ⟶ c)`. -/
@[simps]
def precomposing (a b c : B) : (a ⟶ b) ⥤ (b ⟶ c) ⥤ (a ⟶ c) 
goal: CategoryTheory.Functor (a ⟶ b) (CategoryTheory.Functor (b ⟶ c) (a ⟶ c))

full_name: CategoryTheory.Bicategory.precomposing_obj
code: /-- Precomposition of a 1-morphism as a functor from the category of 1-morphisms `a ⟶ b` into the
category of functors `(b ⟶ c) ⥤ (a ⟶ c)`. -/
@[simps]
def precomposing (a b c : B) : (a ⟶ b) ⥤ (b ⟶ c) ⥤ (a ⟶ c) 
goal: (CategoryTheory.Bicategory.precomposing a b c).obj f = CategoryTheory.Bicategory.precomp c f

full_name: CategoryTheory.Bicategory.postcomp_obj
code: /-- Postcomposition of a 1-morphism as a functor. -/
@[simps]
def postcomp (a : B) (f : b ⟶ c) : (a ⟶ b) ⥤ (a ⟶ c) 
goal: ∀ (x : a ⟶ b), (CategoryTheory.Bicategory.postcomp a f).obj x = CategoryTheory.CategoryStruct.comp x f

full_name: CategoryTheory.Bicategory.postcomp
code: /-- Postcomposition of a 1-morphism as a functor. -/
@[simps]
def postcomp (a : B) (f : b ⟶ c) : (a ⟶ b) ⥤ (a ⟶ c) 
goal: CategoryTheory.Functor (a ⟶ b) (a ⟶ c)

full_name: CategoryTheory.Bicategory.postcomp_map
code: /-- Postcomposition of a 1-morphism as a functor. -/
@[simps]
def postcomp (a : B) (f : b ⟶ c) : (a ⟶ b) ⥤ (a ⟶ c) 
goal: ∀ {X Y : a ⟶ b} (x : X ⟶ Y), (CategoryTheory.Bicategory.postcomp a f).map x = CategoryTheory.Bicategory.whiskerRight x f

full_name: CategoryTheory.Bicategory.postcomposing_map_app
code: /-- Postcomposition of a 1-morphism as a functor from the category of 1-morphisms `b ⟶ c` into the
category of functors `(a ⟶ b) ⥤ (a ⟶ c)`. -/
@[simps]
def postcomposing (a b c : B) : (b ⟶ c) ⥤ (a ⟶ b) ⥤ (a ⟶ c) 
goal: ∀ {X Y : b ⟶ c} (η : X ⟶ Y) (x : a ⟶ b),
  ((CategoryTheory.Bicategory.postcomposing a b c).map η).app x = CategoryTheory.Bicategory.whiskerLeft x η

full_name: CategoryTheory.Bicategory.postcomposing
code: /-- Postcomposition of a 1-morphism as a functor from the category of 1-morphisms `b ⟶ c` into the
category of functors `(a ⟶ b) ⥤ (a ⟶ c)`. -/
@[simps]
def postcomposing (a b c : B) : (b ⟶ c) ⥤ (a ⟶ b) ⥤ (a ⟶ c) 
goal: CategoryTheory.Functor (b ⟶ c) (CategoryTheory.Functor (a ⟶ b) (a ⟶ c))

full_name: CategoryTheory.Bicategory.postcomposing_obj
code: /-- Postcomposition of a 1-morphism as a functor from the category of 1-morphisms `b ⟶ c` into the
category of functors `(a ⟶ b) ⥤ (a ⟶ c)`. -/
@[simps]
def postcomposing (a b c : B) : (b ⟶ c) ⥤ (a ⟶ b) ⥤ (a ⟶ c) 
goal: (CategoryTheory.Bicategory.postcomposing a b c).obj f = CategoryTheory.Bicategory.postcomp a f

full_name: CategoryTheory.Bicategory.associatorNatIsoLeft
code: /-- Left component of the associator as a natural isomorphism. -/
@[simps!]
def associatorNatIsoLeft (a : B) (g : b ⟶ c) (h : c ⟶ d) :
    (postcomposing a ..).obj g ⋙ (postcomposing ..).obj h ≅ (postcomposing ..).obj (g ≫ h) 
goal: ((CategoryTheory.Bicategory.postcomposing a b c).obj g).comp ((CategoryTheory.Bicategory.postcomposing a c d).obj h) ≅
  (CategoryTheory.Bicategory.postcomposing a b d).obj (CategoryTheory.CategoryStruct.comp g h)

full_name: CategoryTheory.Bicategory.associatorNatIsoLeft_inv_app
code: /-- Left component of the associator as a natural isomorphism. -/
@[simps!]
def associatorNatIsoLeft (a : B) (g : b ⟶ c) (h : c ⟶ d) :
    (postcomposing a ..).obj g ⋙ (postcomposing ..).obj h ≅ (postcomposing ..).obj (g ≫ h) 
goal: (CategoryTheory.Bicategory.associatorNatIsoLeft a g h).inv.app X = (CategoryTheory.Bicategory.associator X g h).inv

full_name: CategoryTheory.Bicategory.associatorNatIsoLeft_hom_app
code: /-- Left component of the associator as a natural isomorphism. -/
@[simps!]
def associatorNatIsoLeft (a : B) (g : b ⟶ c) (h : c ⟶ d) :
    (postcomposing a ..).obj g ⋙ (postcomposing ..).obj h ≅ (postcomposing ..).obj (g ≫ h) 
goal: (CategoryTheory.Bicategory.associatorNatIsoLeft a g h).hom.app X = (CategoryTheory.Bicategory.associator X g h).hom

full_name: CategoryTheory.Bicategory.associatorNatIsoMiddle_inv_app
code: /-- Middle component of the associator as a natural isomorphism. -/
@[simps!]
def associatorNatIsoMiddle (f : a ⟶ b) (h : c ⟶ d) :
    (precomposing ..).obj f ⋙ (postcomposing ..).obj h ≅
      (postcomposing ..).obj h ⋙ (precomposing ..).obj f 
goal: (CategoryTheory.Bicategory.associatorNatIsoMiddle f h).inv.app X = (CategoryTheory.Bicategory.associator f X h).inv

full_name: CategoryTheory.Bicategory.associatorNatIsoMiddle_hom_app
code: /-- Middle component of the associator as a natural isomorphism. -/
@[simps!]
def associatorNatIsoMiddle (f : a ⟶ b) (h : c ⟶ d) :
    (precomposing ..).obj f ⋙ (postcomposing ..).obj h ≅
      (postcomposing ..).obj h ⋙ (precomposing ..).obj f 
goal: (CategoryTheory.Bicategory.associatorNatIsoMiddle f h).hom.app X = (CategoryTheory.Bicategory.associator f X h).hom

full_name: CategoryTheory.Bicategory.associatorNatIsoMiddle
code: /-- Middle component of the associator as a natural isomorphism. -/
@[simps!]
def associatorNatIsoMiddle (f : a ⟶ b) (h : c ⟶ d) :
    (precomposing ..).obj f ⋙ (postcomposing ..).obj h ≅
      (postcomposing ..).obj h ⋙ (precomposing ..).obj f 
goal: ((CategoryTheory.Bicategory.precomposing a b c).obj f).comp ((CategoryTheory.Bicategory.postcomposing a c d).obj h) ≅
  ((CategoryTheory.Bicategory.postcomposing b c d).obj h).comp ((CategoryTheory.Bicategory.precomposing a b d).obj f)

full_name: CategoryTheory.Bicategory.associatorNatIsoRight
code: /-- Right component of the associator as a natural isomorphism. -/
@[simps!]
def associatorNatIsoRight (f : a ⟶ b) (g : b ⟶ c) (d : B) :
    (precomposing _ _ d).obj (f ≫ g) ≅ (precomposing ..).obj g ⋙ (precomposing ..).obj f 
goal: (CategoryTheory.Bicategory.precomposing a c d).obj (CategoryTheory.CategoryStruct.comp f g) ≅
  ((CategoryTheory.Bicategory.precomposing b c d).obj g).comp ((CategoryTheory.Bicategory.precomposing a b d).obj f)

full_name: CategoryTheory.Bicategory.associatorNatIsoRight_hom_app
code: /-- Right component of the associator as a natural isomorphism. -/
@[simps!]
def associatorNatIsoRight (f : a ⟶ b) (g : b ⟶ c) (d : B) :
    (precomposing _ _ d).obj (f ≫ g) ≅ (precomposing ..).obj g ⋙ (precomposing ..).obj f 
goal: (CategoryTheory.Bicategory.associatorNatIsoRight f g d).hom.app X = (CategoryTheory.Bicategory.associator f g X).hom

full_name: CategoryTheory.Bicategory.associatorNatIsoRight_inv_app
code: /-- Right component of the associator as a natural isomorphism. -/
@[simps!]
def associatorNatIsoRight (f : a ⟶ b) (g : b ⟶ c) (d : B) :
    (precomposing _ _ d).obj (f ≫ g) ≅ (precomposing ..).obj g ⋙ (precomposing ..).obj f 
goal: (CategoryTheory.Bicategory.associatorNatIsoRight f g d).inv.app X = (CategoryTheory.Bicategory.associator f g X).inv

full_name: CategoryTheory.Bicategory.leftUnitorNatIso_hom_app
code: /-- Left unitor as a natural isomorphism. -/
@[simps!]
def leftUnitorNatIso (a b : B) : (precomposing _ _ b).obj (𝟙 a) ≅ 𝟭 (a ⟶ b) 
goal: (CategoryTheory.Bicategory.leftUnitorNatIso a b).hom.app X = (CategoryTheory.Bicategory.leftUnitor X).hom

full_name: CategoryTheory.Bicategory.leftUnitorNatIso
code: /-- Left unitor as a natural isomorphism. -/
@[simps!]
def leftUnitorNatIso (a b : B) : (precomposing _ _ b).obj (𝟙 a) ≅ 𝟭 (a ⟶ b) 
goal: (CategoryTheory.Bicategory.precomposing a a b).obj (CategoryTheory.CategoryStruct.id a) ≅
  CategoryTheory.Functor.id (a ⟶ b)

full_name: CategoryTheory.Bicategory.leftUnitorNatIso_inv_app
code: /-- Left unitor as a natural isomorphism. -/
@[simps!]
def leftUnitorNatIso (a b : B) : (precomposing _ _ b).obj (𝟙 a) ≅ 𝟭 (a ⟶ b) 
goal: (CategoryTheory.Bicategory.leftUnitorNatIso a b).inv.app X = (CategoryTheory.Bicategory.leftUnitor X).inv

full_name: CategoryTheory.Bicategory.rightUnitorNatIso_hom_app
code: /-- Right unitor as a natural isomorphism. -/
@[simps!]
def rightUnitorNatIso (a b : B) : (postcomposing a _ _).obj (𝟙 b) ≅ 𝟭 (a ⟶ b) 
goal: (CategoryTheory.Bicategory.rightUnitorNatIso a b).hom.app X = (CategoryTheory.Bicategory.rightUnitor X).hom

full_name: CategoryTheory.Bicategory.rightUnitorNatIso_inv_app
code: /-- Right unitor as a natural isomorphism. -/
@[simps!]
def rightUnitorNatIso (a b : B) : (postcomposing a _ _).obj (𝟙 b) ≅ 𝟭 (a ⟶ b) 
goal: (CategoryTheory.Bicategory.rightUnitorNatIso a b).inv.app X = (CategoryTheory.Bicategory.rightUnitor X).inv

full_name: CategoryTheory.Bicategory.rightUnitorNatIso
code: /-- Right unitor as a natural isomorphism. -/
@[simps!]
def rightUnitorNatIso (a b : B) : (postcomposing a _ _).obj (𝟙 b) ≅ 𝟭 (a ⟶ b) 
goal: (CategoryTheory.Bicategory.postcomposing a b b).obj (CategoryTheory.CategoryStruct.id b) ≅
  CategoryTheory.Functor.id (a ⟶ b)