Theorems · Theorem · category theory

CategoryTheory.GradedNatTrans.mk.injEq

∀ {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}
  (app : (X : C) → A.fst ⟶ F.obj X ⟶[V] G.obj X)
  (naturality :
    ∀ (X Y : C),
      CategoryTheory.CategoryStruct.comp (A.snd.β (X ⟶[V] Y)).hom
          (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom (F.map X Y) (app Y))
            (CategoryTheory.eComp V (F.obj X) (F.obj Y) (G.obj Y))) =
        CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom (app X) (G.map X Y))
          (CategoryTheory.eComp V (F.obj X) (G.obj X) (G.obj Y)))
  (app_1 : (X : C) → A.fst ⟶ F.obj X ⟶[V] G.obj X)
  (naturality_1 :
    ∀ (X Y : C),
      CategoryTheory.CategoryStruct.comp (A.snd.β (X ⟶[V] Y)).hom
          (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom (F.map X Y) (app_1 Y))
            (CategoryTheory.eComp V (F.obj X) (F.obj Y) (G.obj Y))) =
        CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom (app_1 X) (G.map X Y))
          (CategoryTheory.eComp V (F.obj X) (G.obj X) (G.obj Y))),
  ({ app := app, naturality := naturality } = { app := app_1, naturality := naturality_1 }) = (app = app_1)

Defined in

Mathlib.CategoryTheory.Enriched.Basic

Cited by

0 results in Mathlib

Foundations

Depth 13 from the axioms · uses propext

Assumes

CategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.EnrichedCategoryCategoryTheory.EnrichedCategory