-- CategoryTheory.Bicategory.pentagon
@[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
Arguments: {B : Type u} [self : CategoryTheory.Bicategory B] {a b c d e : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e)
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