File: Mathlib/CategoryTheory/Bicategory/FunctorBicategory.lean

full_name: CategoryTheory.OplaxNatTrans.whiskerLeft_app
code: /-- Left whiskering of an oplax natural transformation and a modification. -/
@[simps]
def whiskerLeft (η : F ⟶ G) {θ ι : G ⟶ H} (Γ : θ ⟶ ι) : η ≫ θ ⟶ η ≫ ι 
goal: (CategoryTheory.OplaxNatTrans.whiskerLeft η Γ).app a = CategoryTheory.Bicategory.whiskerLeft (η.app a) (Γ.app a)

full_name: CategoryTheory.OplaxNatTrans.whiskerLeft
code: /-- Left whiskering of an oplax natural transformation and a modification. -/
@[simps]
def whiskerLeft (η : F ⟶ G) {θ ι : G ⟶ H} (Γ : θ ⟶ ι) : η ≫ θ ⟶ η ≫ ι 
goal: CategoryTheory.CategoryStruct.comp η θ ⟶ CategoryTheory.CategoryStruct.comp η ι

full_name: CategoryTheory.OplaxNatTrans.whiskerRight
code: /-- Right whiskering of an oplax natural transformation and a modification. -/
@[simps]
def whiskerRight {η θ : F ⟶ G} (Γ : η ⟶ θ) (ι : G ⟶ H) : η ≫ ι ⟶ θ ≫ ι 
goal: CategoryTheory.CategoryStruct.comp η ι ⟶ CategoryTheory.CategoryStruct.comp θ ι

full_name: CategoryTheory.OplaxNatTrans.whiskerRight_app
code: /-- Right whiskering of an oplax natural transformation and a modification. -/
@[simps]
def whiskerRight {η θ : F ⟶ G} (Γ : η ⟶ θ) (ι : G ⟶ H) : η ≫ ι ⟶ θ ≫ ι 
goal: (CategoryTheory.OplaxNatTrans.whiskerRight Γ ι).app a = CategoryTheory.Bicategory.whiskerRight (Γ.app a) (ι.app a)

full_name: CategoryTheory.OplaxNatTrans.associator
code: /-- Associator for the vertical composition of oplax natural transformations. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
def associator (η : F ⟶ G) (θ : G ⟶ H) (ι : H ⟶ I) : (η ≫ θ) ≫ ι ≅ η ≫ θ ≫ ι 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp η θ) ι ≅
  CategoryTheory.CategoryStruct.comp η (CategoryTheory.CategoryStruct.comp θ ι)

full_name: CategoryTheory.OplaxNatTrans.associator_inv_app
code: /-- Associator for the vertical composition of oplax natural transformations. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
def associator (η : F ⟶ G) (θ : G ⟶ H) (ι : H ⟶ I) : (η ≫ θ) ≫ ι ≅ η ≫ θ ≫ ι 
goal: (CategoryTheory.OplaxNatTrans.associator η θ ι).inv.app a =
  (CategoryTheory.Bicategory.associator (η.app a) (θ.app a) (ι.app a)).inv

full_name: CategoryTheory.OplaxNatTrans.associator_hom_app
code: /-- Associator for the vertical composition of oplax natural transformations. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
def associator (η : F ⟶ G) (θ : G ⟶ H) (ι : H ⟶ I) : (η ≫ θ) ≫ ι ≅ η ≫ θ ≫ ι 
goal: (CategoryTheory.OplaxNatTrans.associator η θ ι).hom.app a =
  (CategoryTheory.Bicategory.associator (η.app a) (θ.app a) (ι.app a)).hom

full_name: CategoryTheory.OplaxNatTrans.leftUnitor_hom_app
code: /-- Left unitor for the vertical composition of oplax natural transformations. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
def leftUnitor (η : F ⟶ G) : 𝟙 F ≫ η ≅ η 
goal: (CategoryTheory.OplaxNatTrans.leftUnitor η).hom.app a = (CategoryTheory.Bicategory.leftUnitor (η.app a)).hom

full_name: CategoryTheory.OplaxNatTrans.leftUnitor_inv_app
code: /-- Left unitor for the vertical composition of oplax natural transformations. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
def leftUnitor (η : F ⟶ G) : 𝟙 F ≫ η ≅ η 
goal: (CategoryTheory.OplaxNatTrans.leftUnitor η).inv.app a = (CategoryTheory.Bicategory.leftUnitor (η.app a)).inv

full_name: CategoryTheory.OplaxNatTrans.leftUnitor
code: /-- Left unitor for the vertical composition of oplax natural transformations. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
def leftUnitor (η : F ⟶ G) : 𝟙 F ≫ η ≅ η 
goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.id F) η ≅ η

full_name: CategoryTheory.OplaxNatTrans.rightUnitor_hom_app
code: /-- Right unitor for the vertical composition of oplax natural transformations. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
def rightUnitor (η : F ⟶ G) : η ≫ 𝟙 G ≅ η 
goal: (CategoryTheory.OplaxNatTrans.rightUnitor η).hom.app a = (CategoryTheory.Bicategory.rightUnitor (η.app a)).hom

full_name: CategoryTheory.OplaxNatTrans.rightUnitor_inv_app
code: /-- Right unitor for the vertical composition of oplax natural transformations. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
def rightUnitor (η : F ⟶ G) : η ≫ 𝟙 G ≅ η 
goal: (CategoryTheory.OplaxNatTrans.rightUnitor η).inv.app a = (CategoryTheory.Bicategory.rightUnitor (η.app a)).inv

full_name: CategoryTheory.OplaxNatTrans.rightUnitor
code: /-- Right unitor for the vertical composition of oplax natural transformations. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
def rightUnitor (η : F ⟶ G) : η ≫ 𝟙 G ≅ η 
goal: CategoryTheory.CategoryStruct.comp η (CategoryTheory.CategoryStruct.id G) ≅ η

full_name: CategoryTheory.OplaxFunctor.bicategory_rightUnitor_hom_app
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: (CategoryTheory.Bicategory.rightUnitor η).hom.app a = (CategoryTheory.Bicategory.rightUnitor (η.app a)).hom

full_name: CategoryTheory.OplaxFunctor.bicategory_homCategory_id_app
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: ∀ (a_1 : B), (CategoryTheory.CategoryStruct.id η).app a_1 = CategoryTheory.CategoryStruct.id (η.app a_1)

full_name: CategoryTheory.OplaxFunctor.bicategory_comp_naturality
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: ∀ {X Y Z : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.OplaxNatTrans X Y)
  (θ : CategoryTheory.OplaxNatTrans Y Z) {a b : B} (f : a ⟶ b),
  (CategoryTheory.CategoryStruct.comp η θ).naturality f =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (X.map f) (η.app b) (θ.app b)).inv
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (η.naturality f) (θ.app b))
        (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (η.app a) (Y.map f) (θ.app b)).hom
          (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (η.app a) (θ.naturality f))
            (CategoryTheory.Bicategory.associator (η.app a) (θ.app a) (Z.map f)).inv)))

full_name: CategoryTheory.OplaxFunctor.bicategory_id_naturality
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: (CategoryTheory.CategoryStruct.id F).naturality f =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor (F.map f)).hom
    (CategoryTheory.Bicategory.leftUnitor (F.map f)).inv

full_name: CategoryTheory.OplaxFunctor.bicategory_whiskerLeft_app
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: ∀ (x x_1 : G ⟶ H) (Γ : x ⟶ x_1) (a : B),
  (CategoryTheory.Bicategory.whiskerLeft η Γ).app a = CategoryTheory.Bicategory.whiskerLeft (η.app a) (Γ.app a)

full_name: CategoryTheory.OplaxFunctor.bicategory_associator_hom_app
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: (CategoryTheory.Bicategory.associator η θ ι).hom.app a =
  (CategoryTheory.Bicategory.associator (η.app a) (θ.app a) (ι.app a)).hom

full_name: CategoryTheory.OplaxFunctor.bicategory_associator_inv_app
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: (CategoryTheory.Bicategory.associator η θ ι).inv.app a =
  (CategoryTheory.Bicategory.associator (η.app a) (θ.app a) (ι.app a)).inv

full_name: CategoryTheory.OplaxFunctor.bicategory_homCategory_comp_app
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: ∀ {X Y Z : a ⟶ b} (Γ : CategoryTheory.OplaxNatTrans.Modification X Y)
  (Δ : CategoryTheory.OplaxNatTrans.Modification Y Z) (a_1 : B),
  (CategoryTheory.CategoryStruct.comp Γ Δ).app a_1 = CategoryTheory.CategoryStruct.comp (Γ.app a_1) (Δ.app a_1)

full_name: CategoryTheory.OplaxFunctor.bicategory_rightUnitor_inv_app
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: (CategoryTheory.Bicategory.rightUnitor η).inv.app a = (CategoryTheory.Bicategory.rightUnitor (η.app a)).inv

full_name: CategoryTheory.OplaxFunctor.bicategory_leftUnitor_hom_app
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: (CategoryTheory.Bicategory.leftUnitor η).hom.app a = (CategoryTheory.Bicategory.leftUnitor (η.app a)).hom

full_name: CategoryTheory.OplaxFunctor.bicategory_comp_app
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: ∀ {X Y Z : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.OplaxNatTrans X Y)
  (θ : CategoryTheory.OplaxNatTrans Y Z) (a : B),
  (CategoryTheory.CategoryStruct.comp η θ).app a = CategoryTheory.CategoryStruct.comp (η.app a) (θ.app a)

full_name: CategoryTheory.OplaxFunctor.bicategory_whiskerRight_app
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: ∀ (x x_1 : F ⟶ G) (Γ : x ⟶ x_1) (η : G ⟶ H) (a : B),
  (CategoryTheory.Bicategory.whiskerRight Γ η).app a = CategoryTheory.Bicategory.whiskerRight (Γ.app a) (η.app a)

full_name: CategoryTheory.OplaxFunctor.bicategory
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: CategoryTheory.Bicategory (CategoryTheory.OplaxFunctor B C)

full_name: CategoryTheory.OplaxFunctor.bicategory_leftUnitor_inv_app
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: (CategoryTheory.Bicategory.leftUnitor η).inv.app a = (CategoryTheory.Bicategory.leftUnitor (η.app a)).inv

full_name: CategoryTheory.OplaxFunctor.bicategory_id_app
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: (CategoryTheory.CategoryStruct.id F).app a = CategoryTheory.CategoryStruct.id (F.obj a)

full_name: CategoryTheory.OplaxFunctor.bicategory_Hom
code: /-- A bicategory structure on the oplax functors between bicategories. -/
-- Porting note: verified that projections are correct and changed @[simps] to @[simps!]
@[simps!]
instance OplaxFunctor.bicategory : Bicategory (OplaxFunctor B C) 
goal: (F ⟶ G) = CategoryTheory.OplaxNatTrans F G