Covariant Hom Functor is Continuous

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 $\hom \paren {C, \cdot}: \mathbf C \to \mathbf{Set}$ be the covariant hom functor based at $C$.

Then $\hom \paren {C, \cdot}$ is a continuous functor.