Covariant representables preserve limits
= Covariant representables preserve limits
A <representable functor> $\mathcal C(R,-)$ preserves every small <categorical limit> that exists. A map from $R$ into a limit is exactly a compatible family of maps from $R$ to the diagram objects, and such a family is exactly an element of the corresponding limit of sets. The comparison bijection is the one induced by the limit projections.