Theorems · Theorem · category theory

CategoryTheory.Adjunction.Triple.map_rightToLeft_app_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 : G.Full] [inst_3 : G.Faithful]
  (X : C) {Z : C} (h : G.obj (F.obj X) ⟶ Z),
  CategoryTheory.CategoryStruct.comp (G.map (t.rightToLeft.app X)) h =
    CategoryTheory.CategoryStruct.comp (t.adj₂.counit.app X) (CategoryTheory.CategoryStruct.comp (t.adj₁.unit.app X) h)

For an adjoint triple F ⊣ G ⊣ H where G is fully faithful, the images of the components of the natural transformation H ⟶ F under G are the components of the composition of counit of the second adjunction with the unit of the first adjunction.

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