For which categories is its Yoneda embedding essentially surjective?

Question

For a locally small category $\mathcal{C}$, you can embed $\mathcal{C}$ in the functor category $\mathrm{Set}^\mathcal{C}$ via the functor $X \mapsto \mathrm{Hom}_\mathcal{C}(X,{-})$. This embedding is fully faithful by Yoneda's lemma. But for which $\mathcal{C}$ is this embedding essentially surjective too? Asked another way, when is a category $\mathcal{C}$ equivalent to its functor category $\mathrm{Set}^\mathcal{C}$? Asked yet another way, for which categories $\mathcal{C}$ is every functor $\mathcal{C} \to \mathrm{Set}$ representable?

Answer 1

The Yoneda embedding is never essentially surjective.

To see why, note that the constant empty set functor $C_{\varnothing} : \mathcal{C} \to \mathbf{Set}$ is never representable, since for each $X \in \mathcal{C}$ we have $C_{\varnothing}(X) = \varnothing$ but $\mathrm{id}_X \in \mathrm{Hom}(X,X)$, so that $C_{\varnothing} \ncong \mathrm{Hom}(X,{-})$.