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

Proof: Let $F : \mathbf{A} \rightarrow \mathbf{B}$ and $G: \mathbf{B} \rightarrow \mathbf{A}$ be an adjunction with $F \dashv G$. Let $\eta$ be the unit and $\varepsilon$ the counit. Let $\mathbf{Fix}(GF)$ be the subcategory of $\mathbf{A}$ where the objects are $\setbuilder{A \in \mathbf{A}}{\eta_A \text{ is an isomorphism}}$. For morphisms, let $\mathbf{Fix}(GF)(A, A') = \mathbf{A}(A, A')$, so that $\mathbf{Fix}(GF)$ is a full subcategory. Define dually $\mathbf{Fix}(FG)$, a full subcategory of $\mathbf{B}$. If $A \in \mathbf{Fix}(GF)$, we claim that $F(A) \in \mathbf{Fix}(FG)$. Indeed, by the triangle identity we have $\varepsilon_{F(A)} \circ F(\eta_A) = 1_{F(A)}$. Since $\eta_A$ is an isomorphism, so is $F(\eta_A)$, which means that $\varepsilon_{F(A)}$ is its inverse, an isomorphism as well. Thus, since $\varepsilon_{F(A)}$ is an isomorphism, $F(A) \in \mathbf{Fix}(FG)$. A dual argument using the other triangle identity shows that $B \in \mathbf{Fix}(FG)$ implies $G(B) \in \mathbf{Fix}(GF)$. Thus, $F$ and $G$ restrict to functors $F' : \mathbf{Fix}(GF) \rightarrow \mathbf{Fix}(FG)$ and $G' : \mathbf{Fix}(FG) \rightarrow \mathbf{Fix}(GF)$.

$\eta'$ and $\varepsilon'$, the restrictions of $\eta$ and $\varepsilon$, are trivially natural isomorphisms because each of their components are isomorphisms, by construction. Then, $(F', G', \eta', \varepsilon')$ is an equivalence of categories between $\mathbf{Fix}(GF)$ and $\mathbf{Fix}(FG)$.