Theorems · Theorem · category theory

CategoryTheory.Enriched.FunctorCategory.homEquiv_comp_assoc

∀ (V : Type u₁) [inst : CategoryTheory.Category.{v₁, u₁} V] [inst_1 : CategoryTheory.MonoidalCategory V] {C : Type u₂}
  [inst_2 : CategoryTheory.Category.{v₂, u₂} C] {J : Type u₃} [inst_3 : CategoryTheory.Category.{v₃, u₃} J]
  [inst_4 : CategoryTheory.EnrichedOrdinaryCategory V C] {F₁ F₂ F₃ : CategoryTheory.Functor J C}
  [inst_5 : CategoryTheory.Enriched.FunctorCategory.HasEnrichedHom V F₁ F₂]
  [inst_6 : CategoryTheory.Enriched.FunctorCategory.HasEnrichedHom V F₂ F₃]
  [inst_7 : CategoryTheory.Enriched.FunctorCategory.HasEnrichedHom V F₁ F₃] (f : F₁ ⟶ F₂) (g : F₂ ⟶ F₃) {Z : V}
  (h : CategoryTheory.Enriched.FunctorCategory.enrichedHom V F₁ F₃ ⟶ Z),
  CategoryTheory.CategoryStruct.comp
      ((CategoryTheory.Enriched.FunctorCategory.homEquiv V) (CategoryTheory.CategoryStruct.comp f g)) h =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.MonoidalCategoryStruct.leftUnitor (CategoryTheory.MonoidalCategoryStruct.tensorUnit V)).inv
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.MonoidalCategoryStruct.tensorHom ((CategoryTheory.Enriched.FunctorCategory.homEquiv V) f)
          ((CategoryTheory.Enriched.FunctorCategory.homEquiv V) g))
        (CategoryTheory.CategoryStruct.comp (CategoryTheory.Enriched.FunctorCategory.enrichedComp V F₁ F₂ F₃) h))

Defined in

Mathlib.CategoryTheory.Enriched.FunctorCategory

Cited by

0 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.EnrichedOrdinaryCategoryCategoryTheory.Enriched.FunctorCategory.HasEnrichedHomCategoryTheory.Enriched.FunctorCategory.HasEnrichedHomCategoryTheory.Enriched.FunctorCategory.HasEnrichedHom