**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 .