Theorems · Theorem · category theory

CategoryTheory.Pseudofunctor.StrongTrans.naturality_comp_inv_app_assoc

∀ {B : Type u_1} [inst : CategoryTheory.Bicategory B] {F G : CategoryTheory.Pseudofunctor B CategoryTheory.Cat}
  (α : F ⟶ G) {a b c : B} (f : a ⟶ b) (g : b ⟶ c) (X : ↑(F.obj a)) {Z : ↑(G.obj c)}
  (h : (α.app c).toFunctor.obj ((F.map (CategoryTheory.CategoryStruct.comp f g)).toFunctor.obj X) ⟶ Z),
  CategoryTheory.CategoryStruct.comp ((α.naturality (CategoryTheory.CategoryStruct.comp f g)).inv.toNatTrans.app X) h =
    CategoryTheory.CategoryStruct.comp ((G.mapComp f g).hom.toNatTrans.app ((α.app a).toFunctor.obj X))
      (CategoryTheory.CategoryStruct.comp ((G.map g).toFunctor.map ((α.naturality f).inv.toNatTrans.app X))
        (CategoryTheory.CategoryStruct.comp ((α.naturality g).inv.toNatTrans.app ((F.map f).toFunctor.obj X))
          (CategoryTheory.CategoryStruct.comp ((α.app c).toFunctor.map ((F.mapComp f g).inv.toNatTrans.app X)) h)))

Defined in

Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo

Cited by

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Foundations

Depth 40 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.Bicategory