Theorems · Theorem · category theory

CategoryTheory.LaxMonoidalFunctor.isoOfComponents_hom_hom_app

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.MonoidalCategory C] {D : Type u₂}
  [inst_2 : CategoryTheory.Category.{v₂, u₂} D] [inst_3 : CategoryTheory.MonoidalCategory D]
  {F G : CategoryTheory.LaxMonoidalFunctor C D} (e : (X : C) → F.obj X ≅ G.obj X)
  (naturality :
    autoParam
      (∀ {X Y : C} (f : X ⟶ Y),
        CategoryTheory.CategoryStruct.comp (F.map f) (e Y).hom = CategoryTheory.CategoryStruct.comp (e X).hom (G.map f))
      CategoryTheory.LaxMonoidalFunctor.isoOfComponents._auto_1)
  (unit :
    autoParam
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.LaxMonoidal.ε F.toFunctor)
          (e (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)).hom =
        CategoryTheory.Functor.LaxMonoidal.ε G.toFunctor)
      CategoryTheory.LaxMonoidalFunctor.isoOfComponents._auto_3)
  (tensor :
    autoParam
      (∀ (X Y : C),
        CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.LaxMonoidal.μ F.toFunctor X Y)
            (e (CategoryTheory.MonoidalCategoryStruct.tensorObj X Y)).hom =
          CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom (e X).hom (e Y).hom)
            (CategoryTheory.Functor.LaxMonoidal.μ G.toFunctor X Y))
      CategoryTheory.LaxMonoidalFunctor.isoOfComponents._auto_5)
  (X : C), (CategoryTheory.LaxMonoidalFunctor.isoOfComponents e naturality unit tensor).hom.hom.app X = (e X).hom

Defined in

Mathlib.CategoryTheory.Monoidal.NaturalTransformation

Cited by

1 results in Mathlib

Foundations

Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.MonoidalCategoryCategoryTheory.CategoryCategoryTheory.MonoidalCategory