The hom-set functor is often given as the bifunctor:
$$ \mathbf{Hom}(-,-) : \cal C^{op} \times \cal C \to \mathbf{Set}$$
(This is of course under the assumption that $\cal C$ is locally small.)
Is there a precedent for using a topos other than $\mathbf{Set}$? Such as for instance the category of weak infinity/omega groupoids known from Homotopy Type Theory?