Theorems · Theorem · category theory

CategoryTheory.Functor.isEquivalence_of_isRightAdjoint

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  (G : CategoryTheory.Functor C D) [inst_2 : G.IsRightAdjoint]
  [∀ (X : D), CategoryTheory.IsIso ((CategoryTheory.Adjunction.ofIsRightAdjoint G).unit.app X)]
  [∀ (Y : C), CategoryTheory.IsIso ((CategoryTheory.Adjunction.ofIsRightAdjoint G).counit.app Y)], G.IsEquivalence

If the unit and counit for the adjunction corresponding to a right adjoint functor are (pointwise) isomorphisms, then the functor is an equivalence of categories.

Defined in

Mathlib.CategoryTheory.Adjunction.Basic

Cited by

0 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsRightAdjointCategoryTheory.IsIsoCategoryTheory.IsIso