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 $F$ is fully faithful if and only if $u:\mathrm{Id}_{\mathcal{C}}\to G\circ F$ is isomorphism.