Theorems · Theorem · category theory

CategoryTheory.expComparison_ev

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u'} [inst_1 : CategoryTheory.Category.{v, u'} D]
  [inst_2 : CategoryTheory.CartesianMonoidalCategory C] [inst_3 : CategoryTheory.CartesianMonoidalCategory D]
  (F : CategoryTheory.Functor C D) [inst_4 : CategoryTheory.MonoidalClosed C] [inst_5 : CategoryTheory.MonoidalClosed D]
  [inst_6 : CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) F]
  (A B : C),
  CategoryTheory.CategoryStruct.comp
      (CategoryTheory.MonoidalCategoryStruct.whiskerLeft (F.obj A) ((CategoryTheory.expComparison F A).natTrans.app B))
      ((CategoryTheory.ihom.ev (F.obj A)).app (F.obj B)) =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.inv (CategoryTheory.CartesianMonoidalCategory.prodComparison F A (A ⟹ B)))
      (F.map ((CategoryTheory.ihom.ev A).app B))

Defined in

Mathlib.CategoryTheory.Monoidal.Closed.Functor

Cited by

1 results in Mathlib

Foundations

Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CartesianMonoidalCategoryCategoryTheory.CartesianMonoidalCategoryCategoryTheory.MonoidalClosedCategoryTheory.MonoidalClosedCategoryTheory.Limits.PreservesLimitsOfShape