Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be locally small categories and let $F:\mathcal{C}\to \mathcal{D}$ be a functor. Then $F$ admits a right adjoint if and only if for each $d\in\mathcal{D}$, $\hom_{\mathcal{D}}(F(-),d):\mathcal{C}^{op}\to\mathcal{S}\mathrm{et}$ is representable.