Contravariant Hom Functor maps Colimits to Limits

Theorem
Let $\mathbf{Set}$ be the category of sets.

Let $\mathbf C$ be a locally small category.

Let $C$ be an object of $\mathbf C$, and let $\operatorname{Hom} \left({\cdot, C}\right): \mathbf C \to \mathbf{Set}$ be the contravariant hom functor based at $C$.

Then $\operatorname{Hom} \left({\cdot, C}\right)$ maps every colimit to a limit, in that:


 * $\operatorname{Hom} \left({{\varinjlim \,}_j \,D_j, C}\right) \cong {\varprojlim \,}_j \, \operatorname{Hom} \left({D_j, C}\right)$

for every diagram $D: \mathbf J \to \mathbf C$.