Theorems · Theorem · category theory

CategoryTheory.Equivalence.functor_unitIso_comp

∀ {C : Type u₁} {D : Type u₂} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  (self : C ≌ D) (X : C),
  CategoryTheory.CategoryStruct.comp (self.functor.map (self.unitIso.hom.app X))
      (self.counitIso.hom.app (self.functor.obj X)) =
    CategoryTheory.CategoryStruct.id (self.functor.obj X)

The triangle law for the forwards direction of an equivalence: the unit and counit compose to the identity when whiskered along the forwards direction. We state this as a family of equalities among morphisms instead of an equality of natural transformations to avoid abusing defeq or inserting natural transformations like F ⟶ F𝟭.

Defined in

Mathlib.CategoryTheory.Equivalence

Cited by

3 results in Mathlib

Foundations

Depth 21 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Category