Theorems · Theorem · category theory

CategoryTheory.Adjunction.Triple.leftToRight_app_obj_assoc

∀ {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 : F.Full] [inst_3 : F.Faithful]
  {X Z : D} (h : H.obj (G.obj X) ⟶ Z),
  CategoryTheory.CategoryStruct.comp (t.leftToRight.app (G.obj X)) h =
    CategoryTheory.CategoryStruct.comp (t.adj₁.counit.app X) (CategoryTheory.CategoryStruct.comp (t.adj₂.unit.app X) h)

For an adjoint triple F ⊣ G ⊣ H where F and H are fully faithful, the components of the natural transformation F ⟶ H at G are precisely the components of the natural transformation G ⋙ F ⟶ G ⋙ H obtained from the units and counits of the adjunctions.

Defined in

Mathlib.CategoryTheory.Adjunction.Triple

Cited by

0 results in Mathlib

Foundations

Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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