Theorems · Theorem · category theory

CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.comp_homEquiv_symm

∀ {C : Type u_1} {D : Type u_2} {H : Type u_4} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.Category.{v_4, u_4} H]
  {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} {Y : D}
  (h : E.IsPointwiseLeftKanExtensionAt Y) {Z : H}
  (φ :
    (CategoryTheory.CostructuredArrow.proj L Y).comp F ⟶
      (CategoryTheory.Functor.const (CategoryTheory.CostructuredArrow L Y)).obj Z)
  (g : CategoryTheory.CostructuredArrow L Y),
  CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.hom E).app g.left)
      (CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.right E).map g.hom)
        ((CategoryTheory.Limits.IsColimit.homEquiv h).symm φ)) =
    φ.app g

Defined in

Mathlib.CategoryTheory.Functor.KanExtension.Pointwise

Cited by

1 results in Mathlib

Foundations

Depth 31 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Category