Theorems · Definition · category theory

CategoryTheory.Enriched.FunctorCategory.functorHomEquiv

(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 : CategoryTheory.Functor J C} →
                  [inst_5 : CategoryTheory.Enriched.FunctorCategory.HasFunctorEnrichedHom V F₁ F₂] →
                    [CategoryTheory.Enriched.FunctorCategory.HasEnrichedHom V F₁ F₂] →
                      (F₁ ⟶ F₂) ≃
                        (CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Functor J V) ⟶
                          CategoryTheory.Enriched.FunctorCategory.functorEnrichedHom V F₁ F₂)

Given functors F₁ and F₂ in J ⥤ C, where C is a V-enriched ordinary category, this is the bijection (F₁ ⟶ F₂) ≃ (𝟙_ (J ⥤ V) ⟶ functorEnrichedHom V F₁ F₂).

Defined in

Mathlib.CategoryTheory.Enriched.FunctorCategory

Cited by

5 results in Mathlib

Foundations

Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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