covariant hom-functor preserves small limits

Let be a locally small category Then the covariant hom functor preserves small limits.

Let be a diagram

then consider for the diagram

Suppose is a set with morphism

Then each defines morphismss

where by assumption

Thus by universal property, there exists a unique morphism

Thus let

Then this respects the diagram, as

Furthermore suppose there exists a map making the diagram commute. Then for each it follows, that

makes the diagram commute and thus by uniqueness of , it follows, that

or