Exercise 2.2.11

Theorem: Every adjunction restrictions to an equivalence between (full) subcategories

Proof: Let and be an adjunction with . Let be the unit and the counit. Let be the subcategory of where the objects are . For morphisms, let , so that is a full subcategory. Define dually , a full subcategory of . If , we claim that . Indeed, by the triangle identity we have . Since is an isomorphism, so is , which means that is its inverse, an isomorphism as well. Thus, since is an isomorphism, . A dual argument using the other triangle identity shows that implies . Thus, and restrict to functors and . and , the restrictions of and , are trivially natural isomorphisms because each of their components are isomorphisms, by construction. Then, is an equivalence of categories between and .