Theorems · Theorem · category theory

CategoryTheory.Pseudofunctor.StrongTrans.whiskerRight_naturality_comp

∀ {B : Type u₁} [inst : CategoryTheory.Bicategory B] {C : Type u₂} [inst_1 : CategoryTheory.Bicategory C]
  {F G : CategoryTheory.Pseudofunctor B C} (η : F ⟶ G) {a b c : B} {a' : C} (f : a ⟶ b) (g : b ⟶ c) (h : G.obj c ⟶ a'),
  CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (η.naturality (CategoryTheory.CategoryStruct.comp f g)).hom h)
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Bicategory.associator (η.app a) (G.map (CategoryTheory.CategoryStruct.comp f g)) h).hom
        (CategoryTheory.Bicategory.whiskerLeft (η.app a)
          (CategoryTheory.Bicategory.whiskerRight (G.mapComp f g).hom h))) =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight (F.mapComp f g).hom (η.app c)) h)
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator (F.map f) (F.map g) (η.app c)).hom
          h)
        (CategoryTheory.CategoryStruct.comp
          (CategoryTheory.Bicategory.associator (F.map f) (CategoryTheory.CategoryStruct.comp (F.map g) (η.app c))
              h).hom
          (CategoryTheory.CategoryStruct.comp
            (CategoryTheory.Bicategory.whiskerLeft (F.map f)
              (CategoryTheory.Bicategory.whiskerRight (η.naturality g).hom h))
            (CategoryTheory.CategoryStruct.comp
              (CategoryTheory.Bicategory.associator (F.map f) (CategoryTheory.CategoryStruct.comp (η.app b) (G.map g))
                  h).inv
              (CategoryTheory.CategoryStruct.comp
                (CategoryTheory.Bicategory.whiskerRight
                  (CategoryTheory.Bicategory.associator (F.map f) (η.app b) (G.map g)).inv h)
                (CategoryTheory.CategoryStruct.comp
                  (CategoryTheory.Bicategory.whiskerRight
                    (CategoryTheory.Bicategory.whiskerRight (η.naturality f).hom (G.map g)) h)
                  (CategoryTheory.CategoryStruct.comp
                    (CategoryTheory.Bicategory.whiskerRight
                      (CategoryTheory.Bicategory.associator (η.app a) (G.map f) (G.map g)).hom h)
                    (CategoryTheory.Bicategory.associator (η.app a)
                        (CategoryTheory.CategoryStruct.comp (G.map f) (G.map g)) h).hom)))))))

Defined in

Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo

Cited by

2 results in Mathlib

Foundations

Depth 36 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.BicategoryCategoryTheory.Bicategory