Theorems · Theorem · category theory

CategoryTheory.ExponentiableMorphism.unit_pushforwardComp_hom_assoc

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {I J K : C} (f : I ⟶ J) (g : J ⟶ K)
  [inst_1 : CategoryTheory.ChosenPullbacksAlong f] [inst_2 : CategoryTheory.ChosenPullbacksAlong g]
  [inst_3 : CategoryTheory.ChosenPullbacksAlong (CategoryTheory.CategoryStruct.comp f g)]
  [inst_4 : CategoryTheory.ExponentiableMorphism f] [inst_5 : CategoryTheory.ExponentiableMorphism g]
  [inst_6 : CategoryTheory.ExponentiableMorphism (CategoryTheory.CategoryStruct.comp f g)]
  {Z : CategoryTheory.Functor (CategoryTheory.Over K) (CategoryTheory.Over K)}
  (h :
    (CategoryTheory.ChosenPullbacksAlong.pullback (CategoryTheory.CategoryStruct.comp f g)).comp
        ((CategoryTheory.ExponentiableMorphism.pushforward f).comp
          (CategoryTheory.ExponentiableMorphism.pushforward g)) ⟶
      Z),
  CategoryTheory.CategoryStruct.comp
      (CategoryTheory.ExponentiableMorphism.pullbackPushforwardAdj (CategoryTheory.CategoryStruct.comp f g)).unit
      (CategoryTheory.CategoryStruct.comp
        ((CategoryTheory.ChosenPullbacksAlong.pullback (CategoryTheory.CategoryStruct.comp f g)).whiskerLeft
          (CategoryTheory.ExponentiableMorphism.pushforwardComp f g).hom)
        h) =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.ExponentiableMorphism.pullbackPushforwardAdj (CategoryTheory.CategoryStruct.comp f g)).unit h

Defined in

Mathlib.CategoryTheory.LocallyCartesianClosed.ExponentiableMorphism

Cited by

0 results in Mathlib

Foundations

Depth 43 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.ChosenPullbacksAlongCategoryTheory.ChosenPullbacksAlongCategoryTheory.ChosenPullbacksAlongCategoryTheory.ExponentiableMorphismCategoryTheory.ExponentiableMorphismCategoryTheory.ExponentiableMorphism