Let $\bf C$ be a category, $\mathcal S$ an (elementary) topos.

If $\mathcal S$ is a presheaf category over $\bf D$, then it's easy to see $[\bf C^{\rm op},\, \mathcal S] \cong [(\bf C \times \bf D)^{\rm op},\, \cal{Sets}]$ is still a topos. In more general situations I struggle to see an easy reason for it to be true as well. Thus:

When is the category $[\bf C^{\rm op},\, \mathcal S]$ of contravariant functors from $\bf C$ to $\mathcal S$ a topos?