Theorems · Theorem · category theory

CategoryTheory.CartesianMonoidalCategory.prodComparison_inv_natural_assoc

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.CartesianMonoidalCategory C]
  {D : Type u₁} [inst_2 : CategoryTheory.Category.{v₁, u₁} D] [inst_3 : CategoryTheory.CartesianMonoidalCategory D]
  (F : CategoryTheory.Functor C D) {A B A' B' : C}
  [inst_4 : CategoryTheory.IsIso (CategoryTheory.CartesianMonoidalCategory.prodComparison F A B)] (f : A ⟶ A')
  (g : B ⟶ B') [inst_5 : CategoryTheory.IsIso (CategoryTheory.CartesianMonoidalCategory.prodComparison F A' B')] {Z : D}
  (h : F.obj (CategoryTheory.MonoidalCategoryStruct.tensorObj A' B') ⟶ Z),
  CategoryTheory.CategoryStruct.comp
      (CategoryTheory.inv (CategoryTheory.CartesianMonoidalCategory.prodComparison F A B))
      (CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.MonoidalCategoryStruct.tensorHom f g)) h) =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.tensorHom (F.map f) (F.map g))
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.inv (CategoryTheory.CartesianMonoidalCategory.prodComparison F A' B')) h)

If the product comparison morphism is an iso, its inverse is natural in both argument.

Defined in

Mathlib.CategoryTheory.Monoidal.Cartesian.Basic

Cited by

0 results in Mathlib

Foundations

Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CartesianMonoidalCategoryCategoryTheory.CategoryCategoryTheory.CartesianMonoidalCategoryCategoryTheory.IsIsoCategoryTheory.IsIso