ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Bicategory.LeftExtension.ofCompId_right
@[simps!]
def ofCompId (t : LeftExtension f (g ≫ 𝟙 c)) : LeftExtension f g 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} (t :
  CategoryTheory.Bicategory.LeftExtension f (CategoryTheory.CategoryStruct.comp g (CategoryTheory.CategoryStruct.id c)))
Goal: t.ofCompId.right = t.extension
-- CategoryTheory.Bicategory.LeftExtension.ofCompId_hom
@[simps!]
def ofCompId (t : LeftExtension f (g ≫ 𝟙 c)) : LeftExtension f g 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} (t :
  CategoryTheory.Bicategory.LeftExtension f (CategoryTheory.CategoryStruct.comp g (CategoryTheory.CategoryStruct.id c)))
Goal: t.ofCompId.hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor g).inv t.unit
-- CategoryTheory.Bicategory.LeftExtension.ofCompId_left_as
@[simps!]
def ofCompId (t : LeftExtension f (g ≫ 𝟙 c)) : LeftExtension f g 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} (t :
  CategoryTheory.Bicategory.LeftExtension f (CategoryTheory.CategoryStruct.comp g (CategoryTheory.CategoryStruct.id c)))
Goal: t.ofCompId.left.as = PUnit.unit
-- CategoryTheory.Bicategory.LeftExtension.whisker_extension
@[simp]
theorem whisker_extension (t : LeftExtension f g) {x : B} (h : c ⟶ x) :
    (t.whisker h).extension = t.extension ≫ h 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} (t : CategoryTheory.Bicategory.LeftExtension f g) {x : B} (h : c ⟶ x)
Goal: (t.whisker h).extension = CategoryTheory.CategoryStruct.comp t.extension h
-- CategoryTheory.Bicategory.LeftExtension.whisker_unit
@[simp]
theorem whisker_unit (t : LeftExtension f g) {x : B} (h : c ⟶ x) :
    (t.whisker h).unit = t.unit ▷ h ≫ (α_ f t.extension h).hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} (t : CategoryTheory.Bicategory.LeftExtension f g) {x : B} (h : c ⟶ x)
Goal: (t.whisker h).unit =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight t.unit h)
    (CategoryTheory.Bicategory.associator f t.extension h).hom
-- CategoryTheory.Bicategory.LeftExtension.whiskering_obj
@[simps]
def whiskering {x : B} (h : c ⟶ x) : LeftExtension f g ⥤ LeftExtension f (g ≫ h) 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} {x : B} (h : c ⟶ x) (t : CategoryTheory.Bicategory.LeftExtension f g)
Goal: (CategoryTheory.Bicategory.LeftExtension.whiskering h).obj t = t.whisker h
-- CategoryTheory.Bicategory.LeftExtension.whiskering_map
@[simps]
def whiskering {x : B} (h : c ⟶ x) : LeftExtension f g ⥤ LeftExtension f (g ≫ h) 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} {x : B} (h : c ⟶ x)
Goal: ∀ {X Y : CategoryTheory.Bicategory.LeftExtension f g} (η : X ⟶ Y),
  (CategoryTheory.Bicategory.LeftExtension.whiskering h).map η =
    CategoryTheory.Bicategory.LeftExtension.homMk (CategoryTheory.Bicategory.whiskerRight η.right h) ⋯
-- CategoryTheory.Bicategory.LeftExtension.whiskerIdCancel_right
@[simps! right]
def whiskerIdCancel
    (s : LeftExtension f (g ≫ 𝟙 c)) {t : LeftExtension f g} (τ : s ⟶ t.whisker (𝟙 c)) :
    s.ofCompId ⟶ t 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} (s :
  CategoryTheory.Bicategory.LeftExtension f (CategoryTheory.CategoryStruct.comp g (CategoryTheory.CategoryStruct.id c))) {t : CategoryTheory.Bicategory.LeftExtension f g} (τ : s ⟶ t.whisker (CategoryTheory.CategoryStruct.id c))
Goal: (s.whiskerIdCancel τ).right =
  CategoryTheory.CategoryStruct.comp τ.right (CategoryTheory.Bicategory.rightUnitor t.extension).hom
-- CategoryTheory.Bicategory.LeftExtension.whiskerHom_right
@[simps! right]
def whiskerHom (i : s ⟶ t) {x : B} (h : c ⟶ x) :
    s.whisker h ⟶ t.whisker h 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} {s t : CategoryTheory.Bicategory.LeftExtension f g} (i : s ⟶ t) {x : B} (h : c ⟶ x)
Goal: (CategoryTheory.Bicategory.LeftExtension.whiskerHom i h).right = CategoryTheory.Bicategory.whiskerRight i.right h
-- CategoryTheory.Bicategory.LeftExtension.whiskerOfCompIdIsoSelf_inv_right
@[simps! hom_right inv_right]
def whiskerOfCompIdIsoSelf (t : LeftExtension f g) : (t.whisker (𝟙 c)).ofCompId ≅ t 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} (t : CategoryTheory.Bicategory.LeftExtension f g)
Goal: t.whiskerOfCompIdIsoSelf.inv.right = (CategoryTheory.Bicategory.rightUnitor t.extension).inv
-- CategoryTheory.Bicategory.LeftExtension.whiskerOfCompIdIsoSelf_hom_right
@[simps! hom_right inv_right]
def whiskerOfCompIdIsoSelf (t : LeftExtension f g) : (t.whisker (𝟙 c)).ofCompId ≅ t 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} (t : CategoryTheory.Bicategory.LeftExtension f g)
Goal: t.whiskerOfCompIdIsoSelf.hom.right = (CategoryTheory.Bicategory.rightUnitor t.extension).hom
-- CategoryTheory.Bicategory.LeftLift.w
@[reassoc (attr := simp)]
theorem w (h : s ⟶ t) : s.unit ≫ h.right ▷ f = t.unit 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} {s t : CategoryTheory.Bicategory.LeftLift f g} (h : s ⟶ t)
Goal: CategoryTheory.CategoryStruct.comp s.unit (CategoryTheory.Bicategory.whiskerRight h.right f) = t.unit
-- CategoryTheory.Bicategory.LeftLift.ofIdComp_right
@[simps!]
def ofIdComp (t : LeftLift f (𝟙 c ≫ g)) : LeftLift f g 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} (t : CategoryTheory.Bicategory.LeftLift f (CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id c) g))
Goal: t.ofIdComp.right = t.lift
-- CategoryTheory.Bicategory.LeftLift.ofIdComp_hom
@[simps!]
def ofIdComp (t : LeftLift f (𝟙 c ≫ g)) : LeftLift f g 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} (t : CategoryTheory.Bicategory.LeftLift f (CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id c) g))
Goal: t.ofIdComp.hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor g).inv t.unit
-- CategoryTheory.Bicategory.LeftLift.ofIdComp_left_as
@[simps!]
def ofIdComp (t : LeftLift f (𝟙 c ≫ g)) : LeftLift f g 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} (t : CategoryTheory.Bicategory.LeftLift f (CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id c) g))
Goal: t.ofIdComp.left.as = PUnit.unit
-- CategoryTheory.Bicategory.LeftLift.whisker_lift
@[simp]
theorem whisker_lift (t : LeftLift f g) {x : B} (h : x ⟶ c) :
    (t.whisker h).lift = h ≫ t.lift 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} (t : CategoryTheory.Bicategory.LeftLift f g) {x : B} (h : x ⟶ c)
Goal: (t.whisker h).lift = CategoryTheory.CategoryStruct.comp h t.lift
-- CategoryTheory.Bicategory.LeftLift.whisker_unit
@[simp]
theorem whisker_unit (t : LeftLift f g) {x : B} (h : x ⟶ c) :
    (t.whisker h).unit = h ◁ t.unit ≫ (α_ h t.lift f).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} (t : CategoryTheory.Bicategory.LeftLift f g) {x : B} (h : x ⟶ c)
Goal: (t.whisker h).unit =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft h t.unit)
    (CategoryTheory.Bicategory.associator h t.lift f).inv
-- CategoryTheory.Bicategory.LeftLift.whiskering_map
@[simps]
def whiskering {x : B} (h : x ⟶ c) : LeftLift f g ⥤ LeftLift f (h ≫ g) 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} {x : B} (h : x ⟶ c)
Goal: ∀ {X Y : CategoryTheory.Bicategory.LeftLift f g} (η : X ⟶ Y),
  (CategoryTheory.Bicategory.LeftLift.whiskering h).map η =
    CategoryTheory.Bicategory.LeftLift.homMk (CategoryTheory.Bicategory.whiskerLeft h η.right) ⋯
-- CategoryTheory.Bicategory.LeftLift.whiskering_obj
@[simps]
def whiskering {x : B} (h : x ⟶ c) : LeftLift f g ⥤ LeftLift f (h ≫ g) 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} {x : B} (h : x ⟶ c) (t : CategoryTheory.Bicategory.LeftLift f g)
Goal: (CategoryTheory.Bicategory.LeftLift.whiskering h).obj t = t.whisker h
-- CategoryTheory.Bicategory.LeftLift.whiskerIdCancel_right
@[simps! right]
def whiskerIdCancel
    (s : LeftLift f (𝟙 c ≫ g)) {t : LeftLift f g} (τ : s ⟶ t.whisker (𝟙 c)) :
    s.ofIdComp ⟶ t 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} (s : CategoryTheory.Bicategory.LeftLift f (CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id c) g)) {t : CategoryTheory.Bicategory.LeftLift f g} (τ : s ⟶ t.whisker (CategoryTheory.CategoryStruct.id c))
Goal: (s.whiskerIdCancel τ).right =
  CategoryTheory.CategoryStruct.comp τ.right (CategoryTheory.Bicategory.leftUnitor t.lift).hom
-- CategoryTheory.Bicategory.LeftLift.whiskerHom_right
@[simps! right]
def whiskerHom (i : s ⟶ t) {x : B} (h : x ⟶ c) :
    s.whisker h ⟶ t.whisker h 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} {s t : CategoryTheory.Bicategory.LeftLift f g} (i : s ⟶ t) {x : B} (h : x ⟶ c)
Goal: (CategoryTheory.Bicategory.LeftLift.whiskerHom i h).right = CategoryTheory.Bicategory.whiskerLeft h i.right
-- CategoryTheory.Bicategory.LeftLift.whiskerOfIdCompIsoSelf_hom_right
@[simps! hom_right inv_right]
def whiskerOfIdCompIsoSelf (t : LeftLift f g) : (t.whisker (𝟙 c)).ofIdComp ≅ t 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} (t : CategoryTheory.Bicategory.LeftLift f g)
Goal: t.whiskerOfIdCompIsoSelf.hom.right = (CategoryTheory.Bicategory.leftUnitor t.lift).hom
-- CategoryTheory.Bicategory.LeftLift.whiskerOfIdCompIsoSelf_inv_right
@[simps! hom_right inv_right]
def whiskerOfIdCompIsoSelf (t : LeftLift f g) : (t.whisker (𝟙 c)).ofIdComp ≅ t 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} (t : CategoryTheory.Bicategory.LeftLift f g)
Goal: t.whiskerOfIdCompIsoSelf.inv.right = (CategoryTheory.Bicategory.leftUnitor t.lift).inv
-- CategoryTheory.Bicategory.RightExtension.w
@[reassoc (attr := simp)]
theorem w {s t : RightExtension f g} (η : s ⟶ t) :
    f ◁ η.left ≫ t.counit = s.counit 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} {s t : CategoryTheory.Bicategory.RightExtension f g} (η : s ⟶ t)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft f η.left) t.counit = s.counit
-- CategoryTheory.Bicategory.RightLift.w
@[reassoc (attr := simp)]
theorem w {s t : RightLift f g} (h : s ⟶ t) :
    h.left ▷ f ≫ t.counit = s.counit 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : b ⟶ a} {g : c ⟶ a} {s t : CategoryTheory.Bicategory.RightLift f g} (h : s ⟶ t)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight h.left f) t.counit = s.counit
-- CategoryTheory.LaxFunctor.map₂_rightUnitor
structure LaxFunctor (B: Type u₁) [Bicategory.{w₁, v₁} B] (C : Type u₂) [Bicategory.{w₂, v₂} C]
    extends PrelaxFunctor B C where
    mapId (a : B) : 𝟙 (obj a) ⟶ map (𝟙 a)
    mapComp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : map f ≫ map g ⟶ map (f ≫ g)
    mapComp_naturality_left :
    ∀ {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c),
      mapComp f g ≫ map₂ (η ▷ g) = map₂ η ▷ map g ≫ mapComp f' g:= by aesop_cat
    mapComp_naturality_right :
    ∀ {a b c : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g'),
     mapComp f g ≫ map₂ (f ◁ η) = map f ◁ map₂ η ≫ mapComp f g' := by aesop_cat
    map₂_associator :
    ∀ {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d),
      mapComp f g ▷ map h ≫ mapComp (f ≫ g) h ≫ map₂ (α_ f g h).hom =
      (α_ (map f) (map g) (map h)).hom ≫ map f ◁ mapComp g h ≫ mapComp f (g ≫ h) := by aesop_cat
    map₂_leftUnitor :
    ∀ {a b : B} (f : a ⟶ b),
      map₂ (λ_ f).inv = (λ_ (map f)).inv ≫ mapId a ▷ map f ≫ mapComp (𝟙 a) f := by aesop_cat
    map₂_rightUnitor :
    ∀ {a b : B} (f : a ⟶ b),
      map₂ (ρ_ f).inv = (ρ_ (map f)).inv ≫ map f ◁ mapId b ≫ mapComp f (𝟙 b) := by aesop_cat
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (self : CategoryTheory.LaxFunctor B C) {a b : B} (f : a ⟶ b)
Goal: self.map₂ (CategoryTheory.Bicategory.rightUnitor f).inv =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor (self.map f)).inv
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (self.map f) (self.mapId b))
      (self.mapComp f (CategoryTheory.CategoryStruct.id b)))
-- CategoryTheory.LaxFunctor.mapComp_naturality_left
structure LaxFunctor (B: Type u₁) [Bicategory.{w₁, v₁} B] (C : Type u₂) [Bicategory.{w₂, v₂} C]
    extends PrelaxFunctor B C where
    mapId (a : B) : 𝟙 (obj a) ⟶ map (𝟙 a)
    mapComp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : map f ≫ map g ⟶ map (f ≫ g)
    mapComp_naturality_left :
    ∀ {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c),
      mapComp f g ≫ map₂ (η ▷ g) = map₂ η ▷ map g ≫ mapComp f' g:= by aesop_cat
    mapComp_naturality_right :
    ∀ {a b c : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g'),
     mapComp f g ≫ map₂ (f ◁ η) = map f ◁ map₂ η ≫ mapComp f g' := by aesop_cat
    map₂_associator :
    ∀ {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d),
      mapComp f g ▷ map h ≫ mapComp (f ≫ g) h ≫ map₂ (α_ f g h).hom =
      (α_ (map f) (map g) (map h)).hom ≫ map f ◁ mapComp g h ≫ mapComp f (g ≫ h) := by aesop_cat
    map₂_leftUnitor :
    ∀ {a b : B} (f : a ⟶ b),
      map₂ (λ_ f).inv = (λ_ (map f)).inv ≫ mapId a ▷ map f ≫ mapComp (𝟙 a) f := by aesop_cat
    map₂_rightUnitor :
    ∀ {a b : B} (f : a ⟶ b),
      map₂ (ρ_ f).inv = (ρ_ (map f)).inv ≫ map f ◁ mapId b ≫ mapComp f (𝟙 b) := by aesop_cat
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (self : CategoryTheory.LaxFunctor B C) {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c)
Goal: CategoryTheory.CategoryStruct.comp (self.mapComp f g) (self.map₂ (CategoryTheory.Bicategory.whiskerRight η g)) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (self.map₂ η) (self.map g))
    (self.mapComp f' g)
-- CategoryTheory.LaxFunctor.map₂_associator
structure LaxFunctor (B: Type u₁) [Bicategory.{w₁, v₁} B] (C : Type u₂) [Bicategory.{w₂, v₂} C]
    extends PrelaxFunctor B C where
    mapId (a : B) : 𝟙 (obj a) ⟶ map (𝟙 a)
    mapComp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : map f ≫ map g ⟶ map (f ≫ g)
    mapComp_naturality_left :
    ∀ {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c),
      mapComp f g ≫ map₂ (η ▷ g) = map₂ η ▷ map g ≫ mapComp f' g:= by aesop_cat
    mapComp_naturality_right :
    ∀ {a b c : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g'),
     mapComp f g ≫ map₂ (f ◁ η) = map f ◁ map₂ η ≫ mapComp f g' := by aesop_cat
    map₂_associator :
    ∀ {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d),
      mapComp f g ▷ map h ≫ mapComp (f ≫ g) h ≫ map₂ (α_ f g h).hom =
      (α_ (map f) (map g) (map h)).hom ≫ map f ◁ mapComp g h ≫ mapComp f (g ≫ h) := by aesop_cat
    map₂_leftUnitor :
    ∀ {a b : B} (f : a ⟶ b),
      map₂ (λ_ f).inv = (λ_ (map f)).inv ≫ mapId a ▷ map f ≫ mapComp (𝟙 a) f := by aesop_cat
    map₂_rightUnitor :
    ∀ {a b : B} (f : a ⟶ b),
      map₂ (ρ_ f).inv = (ρ_ (map f)).inv ≫ map f ◁ mapId b ≫ mapComp f (𝟙 b) := by aesop_cat
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (self : CategoryTheory.LaxFunctor B C) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (self.mapComp f g) (self.map h))
    (CategoryTheory.CategoryStruct.comp (self.mapComp (CategoryTheory.CategoryStruct.comp f g) h)
      (self.map₂ (CategoryTheory.Bicategory.associator f g h).hom)) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (self.map f) (self.map g) (self.map h)).hom
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (self.map f) (self.mapComp g h))
      (self.mapComp f (CategoryTheory.CategoryStruct.comp g h)))
-- CategoryTheory.LaxFunctor.map₂_leftUnitor
structure LaxFunctor (B: Type u₁) [Bicategory.{w₁, v₁} B] (C : Type u₂) [Bicategory.{w₂, v₂} C]
    extends PrelaxFunctor B C where
    mapId (a : B) : 𝟙 (obj a) ⟶ map (𝟙 a)
    mapComp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : map f ≫ map g ⟶ map (f ≫ g)
    mapComp_naturality_left :
    ∀ {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c),
      mapComp f g ≫ map₂ (η ▷ g) = map₂ η ▷ map g ≫ mapComp f' g:= by aesop_cat
    mapComp_naturality_right :
    ∀ {a b c : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g'),
     mapComp f g ≫ map₂ (f ◁ η) = map f ◁ map₂ η ≫ mapComp f g' := by aesop_cat
    map₂_associator :
    ∀ {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d),
      mapComp f g ▷ map h ≫ mapComp (f ≫ g) h ≫ map₂ (α_ f g h).hom =
      (α_ (map f) (map g) (map h)).hom ≫ map f ◁ mapComp g h ≫ mapComp f (g ≫ h) := by aesop_cat
    map₂_leftUnitor :
    ∀ {a b : B} (f : a ⟶ b),
      map₂ (λ_ f).inv = (λ_ (map f)).inv ≫ mapId a ▷ map f ≫ mapComp (𝟙 a) f := by aesop_cat
    map₂_rightUnitor :
    ∀ {a b : B} (f : a ⟶ b),
      map₂ (ρ_ f).inv = (ρ_ (map f)).inv ≫ map f ◁ mapId b ≫ mapComp f (𝟙 b) := by aesop_cat
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (self : CategoryTheory.LaxFunctor B C) {a b : B} (f : a ⟶ b)
Goal: self.map₂ (CategoryTheory.Bicategory.leftUnitor f).inv =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor (self.map f)).inv
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (self.mapId a) (self.map f))
      (self.mapComp (CategoryTheory.CategoryStruct.id a) f))
-- CategoryTheory.LaxFunctor.mapComp_naturality_right
structure LaxFunctor (B: Type u₁) [Bicategory.{w₁, v₁} B] (C : Type u₂) [Bicategory.{w₂, v₂} C]
    extends PrelaxFunctor B C where
    mapId (a : B) : 𝟙 (obj a) ⟶ map (𝟙 a)
    mapComp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : map f ≫ map g ⟶ map (f ≫ g)
    mapComp_naturality_left :
    ∀ {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c),
      mapComp f g ≫ map₂ (η ▷ g) = map₂ η ▷ map g ≫ mapComp f' g:= by aesop_cat
    mapComp_naturality_right :
    ∀ {a b c : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g'),
     mapComp f g ≫ map₂ (f ◁ η) = map f ◁ map₂ η ≫ mapComp f g' := by aesop_cat
    map₂_associator :
    ∀ {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d),
      mapComp f g ▷ map h ≫ mapComp (f ≫ g) h ≫ map₂ (α_ f g h).hom =
      (α_ (map f) (map g) (map h)).hom ≫ map f ◁ mapComp g h ≫ mapComp f (g ≫ h) := by aesop_cat
    map₂_leftUnitor :
    ∀ {a b : B} (f : a ⟶ b),
      map₂ (λ_ f).inv = (λ_ (map f)).inv ≫ mapId a ▷ map f ≫ mapComp (𝟙 a) f := by aesop_cat
    map₂_rightUnitor :
    ∀ {a b : B} (f : a ⟶ b),
      map₂ (ρ_ f).inv = (ρ_ (map f)).inv ≫ map f ◁ mapId b ≫ mapComp f (𝟙 b) := by aesop_cat
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (self : CategoryTheory.LaxFunctor B C) {a b c : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g')
Goal: CategoryTheory.CategoryStruct.comp (self.mapComp f g) (self.map₂ (CategoryTheory.Bicategory.whiskerLeft f η)) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (self.map f) (self.map₂ η))
    (self.mapComp f g')
-- CategoryTheory.LaxFunctor.mapComp_assoc_left
@[reassoc]
lemma mapComp_assoc_left {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) :
    F.mapComp f g ▷ F.map h ≫ F.mapComp (f ≫ g) h = (α_ (F.map f) (F.map g) (F.map h)).hom ≫
      F.map f ◁ F.mapComp g h ≫ F.mapComp f (g ≫ h) ≫ F.map₂ (α_ f g h).inv 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F : CategoryTheory.LaxFunctor B C) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (F.mapComp f g) (F.map h))
    (F.mapComp (CategoryTheory.CategoryStruct.comp f g) h) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (F.map f) (F.map g) (F.map h)).hom
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (F.map f) (F.mapComp g h))
      (CategoryTheory.CategoryStruct.comp (F.mapComp f (CategoryTheory.CategoryStruct.comp g h))
        (F.map₂ (CategoryTheory.Bicategory.associator f g h).inv)))
-- CategoryTheory.LaxFunctor.mapComp_assoc_right
@[reassoc]
lemma mapComp_assoc_right {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) :
    F.map f ◁ F.mapComp g h ≫ F.mapComp f (g ≫ h) =
      (α_ (F.map f) (F.map g) (F.map h)).inv ≫ F.mapComp f g ▷ F.map h ≫
        F.mapComp (f ≫ g) h ≫ F.map₂ (α_ f g h).hom 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F : CategoryTheory.LaxFunctor B C) {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (F.map f) (F.mapComp g h))
    (F.mapComp f (CategoryTheory.CategoryStruct.comp g h)) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (F.map f) (F.map g) (F.map h)).inv
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (F.mapComp f g) (F.map h))
      (CategoryTheory.CategoryStruct.comp (F.mapComp (CategoryTheory.CategoryStruct.comp f g) h)
        (F.map₂ (CategoryTheory.Bicategory.associator f g h).hom)))
-- CategoryTheory.LaxFunctor.map₂_leftUnitor_hom
@[reassoc]
lemma map₂_leftUnitor_hom {a b : B} (f : a ⟶ b) :
    (λ_ (F.map f)).hom = F.mapId a ▷ F.map f ≫ F.mapComp (𝟙 a) f ≫ F.map₂ (λ_ f).hom 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F : CategoryTheory.LaxFunctor B C) {a b : B} (f : a ⟶ b)
Goal: (CategoryTheory.Bicategory.leftUnitor (F.map f)).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (F.mapId a) (F.map f))
    (CategoryTheory.CategoryStruct.comp (F.mapComp (CategoryTheory.CategoryStruct.id a) f)
      (F.map₂ (CategoryTheory.Bicategory.leftUnitor f).hom))
-- CategoryTheory.LaxFunctor.map₂_rightUnitor_hom
@[reassoc]
lemma map₂_rightUnitor_hom {a b : B} (f : a ⟶ b) :
    (ρ_ (F.map f)).hom = F.map f ◁ F.mapId b ≫ F.mapComp f (𝟙 b) ≫ F.map₂ (ρ_ f).hom 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F : CategoryTheory.LaxFunctor B C) {a b : B} (f : a ⟶ b)
Goal: (CategoryTheory.Bicategory.rightUnitor (F.map f)).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (F.map f) (F.mapId b))
    (CategoryTheory.CategoryStruct.comp (F.mapComp f (CategoryTheory.CategoryStruct.id b))
      (F.map₂ (CategoryTheory.Bicategory.rightUnitor f).hom))
-- CategoryTheory.LaxFunctor.id_toPrelaxFunctor
@[simps]
def id (B : Type u₁) [Bicategory.{w₁, v₁} B] : LaxFunctor B B 
Arguments: (B : Type u₁) [CategoryTheory.Bicategory B]
Goal: (CategoryTheory.LaxFunctor.id B).toPrelaxFunctor = CategoryTheory.PrelaxFunctor.id B
-- CategoryTheory.LaxFunctor.id_mapComp
@[simps]
def id (B : Type u₁) [Bicategory.{w₁, v₁} B] : LaxFunctor B B 
Arguments: (B : Type u₁) [CategoryTheory.Bicategory B]
Goal: ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c),
  (CategoryTheory.LaxFunctor.id B).mapComp f g =
    CategoryTheory.CategoryStruct.id (CategoryTheory.CategoryStruct.comp f g)
-- CategoryTheory.LaxFunctor.id_mapId
@[simps]
def id (B : Type u₁) [Bicategory.{w₁, v₁} B] : LaxFunctor B B 
Arguments: (B : Type u₁) [CategoryTheory.Bicategory B] (a : B)
Goal: (CategoryTheory.LaxFunctor.id B).mapId a = CategoryTheory.CategoryStruct.id (CategoryTheory.CategoryStruct.id a)
-- CategoryTheory.LaxFunctor.comp_toPrelaxFunctor
@[simps]
def comp {D : Type u₃} [Bicategory.{w₃, v₃} D] (F : LaxFunctor B C) (G : LaxFunctor C D) :
    LaxFunctor B D 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {D : Type u₃} [CategoryTheory.Bicategory D] (F : CategoryTheory.LaxFunctor B C) (G : CategoryTheory.LaxFunctor C D)
Goal: (F.comp G).toPrelaxFunctor = F.comp G.toPrelaxFunctor