Theorems · Theorem · category theory

CategoryTheory.Limits.prodComparison_inv_natural_assoc

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{w, u₂} D]
  (F : CategoryTheory.Functor C D) {A A' B B' : C} [inst_2 : CategoryTheory.Limits.HasBinaryProduct A B]
  [inst_3 : CategoryTheory.Limits.HasBinaryProduct A' B']
  [inst_4 : CategoryTheory.Limits.HasBinaryProduct (F.obj A) (F.obj B)]
  [inst_5 : CategoryTheory.Limits.HasBinaryProduct (F.obj A') (F.obj B')] (f : A ⟶ A') (g : B ⟶ B')
  [inst_6 : CategoryTheory.IsIso (CategoryTheory.Limits.prodComparison F A B)]
  [inst_7 : CategoryTheory.IsIso (CategoryTheory.Limits.prodComparison F A' B')] {Z : D} (h : F.obj (A' ⨯ B') ⟶ Z),
  CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (CategoryTheory.Limits.prodComparison F A B))
      (CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.Limits.prod.map f g)) h) =
    CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.prod.map (F.map f) (F.map g))
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (CategoryTheory.Limits.prodComparison F A' B')) h)

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

Defined in

Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts

Cited by

0 results in Mathlib

Foundations

Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasBinaryProductCategoryTheory.Limits.HasBinaryProductCategoryTheory.Limits.HasBinaryProductCategoryTheory.Limits.HasBinaryProductCategoryTheory.IsIsoCategoryTheory.IsIso