Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F:\mathcal{C}\to \mathcal{D}$ be a functor that admits a right adjoint $G$. Then $G$ is an equivalence of categories if and only if $F$ is fully faithful and $G$ is conservative.
Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F:\mathcal{C}\to \mathcal{D}$ be a functor that admits a right adjoint $G$. Then $G$ is an equivalence of categories if and only if $F$ is fully faithful and $G$ is conservative.