 Hom-set detection of categorical limits

A categorical cone $L\to E_i$ is a categorical limit precisely when $\mathcal{C}(C,L)\to {\lim }_i\mathcal{C}(C,E_i)$ is a bijection for every object $C$. This is exactly the universal property expressed as a test on hom-sets. Consequently a functor preserves existing categorical limits if every composite with a covariant representable functor does.