Theorems · Theorem · category theory

CategoryTheory.GradedNatTrans.naturality_assoc

∀ {V : Type v} [inst : CategoryTheory.Category.{w, v} V] [inst_1 : CategoryTheory.MonoidalCategory V] {C : Type u₁}
  [inst_2 : CategoryTheory.EnrichedCategory V C] {D : Type u₂} [inst_3 : CategoryTheory.EnrichedCategory V D]
  {A : CategoryTheory.Center V} {F G : CategoryTheory.EnrichedFunctor V C D}
  (self : CategoryTheory.GradedNatTrans A F G) (X Y : C) {Z : V} (h : (F.obj X ⟶[V] G.obj Y) ⟶ Z),
  CategoryTheory.CategoryStruct.comp (A.snd.β (X ⟶[V] Y)).hom
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom (F.map X Y) (self.app Y))
        (CategoryTheory.CategoryStruct.comp (CategoryTheory.eComp V (F.obj X) (F.obj Y) (G.obj Y)) h)) =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom (self.app X) (G.map X Y))
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.eComp V (F.obj X) (G.obj X) (G.obj Y)) h)

app is a natural transformation.

Defined in

Mathlib.CategoryTheory.Enriched.Basic

Cited by

0 results in Mathlib

Foundations

Depth 7 from the axioms · uses Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.EnrichedCategoryCategoryTheory.EnrichedCategory