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 $\operatorname{Hom} \left({C, \cdot}\right): \mathbf C \to \mathbf{Set}$ be the covariant hom functor based at $C$.

Then $\operatorname{Hom} \left({C, \cdot}\right)$ is a continuous functor.