Theorems · Theorem · category theory

CategoryTheory.Adjunction.Triple.rightToLeft_eq_counits

∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C}
  {H : CategoryTheory.Functor C D} (t : CategoryTheory.Adjunction.Triple F G H) [inst_2 : G.Full] [inst_3 : G.Faithful],
  t.rightToLeft =
    CategoryTheory.CategoryStruct.comp H.rightUnitor.inv
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (H.whiskerLeft t.adj₁.counit))
        (CategoryTheory.CategoryStruct.comp (H.associator G F).inv
          (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight t.adj₂.counit F) F.leftUnitor.hom)))

The natural transformation H ⟶ F for an adjoint triple F ⊣ G ⊣ H with G fully faithful is also equal to the inverse of the whiskered counit H ⋙ G ⋙ F ⟶ H of the first adjunction followed by the whiskered counit H ⋙ G ⋙ F ⟶ F of the second.

Defined in

Mathlib.CategoryTheory.Adjunction.Triple

Cited by

2 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.FullCategoryTheory.Functor.Faithful