Covariant representables are projective

For a locally small category $\mathcal{C}$, every representable functor $\mathcal{C}(A,-)$ is a projective object in a category in $[\mathcal{C},\mathbf{Set}]$. The Yoneda lemma identifies maps from this functor with elements at $A$, and the pointwise epimorphism in a functor category criterion lets such an element be lifted through any epimorphism.