Condition for an equivalence of functor categories to imply an equivalence of categories

Question

Given small categories $\mathcal{C}$ and $\mathcal{D}$, we have that $[\mathcal{C}^\text{op},\textbf{Set}]\simeq[\mathcal{D}^\text{op},\textbf{Set}]$ if and only if the Cauchy-completions of $\mathcal{C}$ and $\mathcal{D}$ are equivalent. If we replace $\textbf{Set}$ with an arbitrary topos $\mathscr{S}$, is there an equivalent condition (dependent on $\mathscr{S}$) for their functor categories into $\mathscr{S}$ to be equivalent?

The Cauchy completeness of $\mathcal{C}$ can be defined (for example in ncatlab) as the full subcategory of the category of presheaves on $\mathcal{C}$ on the retracts of representable functors. "Representable functors" only exist from $\mathcal{C}^\text{op}$ to $\textbf{Set}$ because $\mathcal{C}$ is enriched over $\textbf{Set}$, so does there need to be a similar condition on $\mathscr{S}$?

(This is a duplicate of my question incorrectly posted on stack exchange).

Answer 1

You can find this theoerm as Proposition 5.28 in Kelly's Basic concepts of enriched category theory. I guess the only extra condition you need is that the base for enrichment forms a topos. They are all cartesian so this is not a big issue although maybe what you want is a different monoidal structure on the topos.

Answer 2

As AT0 said, there is an analogous theorem in Kelly's book for categories enriched over any base. So this will apply as soon as $C$ and $D$ are enriched over $\mathcal{S}$. But if $\mathcal{S}$ is a Grothendieck topos, then the inverse image of its global sections geometric morphism is (in particular) a finite-product-preserving functor $\mathbf{Set} \to \mathcal{S}$, hence it takes ordinary $\mathbf{Set}$-enriched categories to $\mathcal{S}$-enriched ones if applied homwise. So that's a way you could apply this result to ordinary small categories. You'll need to do a little work to identify the relevant $\mathcal{S}$-notion of "Cauchy complete", however.

Another generalization is that the standard theorem is constructive, hence applies internally in any topos. Thus, if $C$ and $D$ are internal categories in $\mathcal{S}$, the same result holds when considering an equivalence of $\mathcal{S}$-indexed presheaf categories and an internal notion of "Cauchy complete". Again, the finitely continuous functor $\mathbf{Set} \to \mathcal{S}$ takes ordinary small categories (which are internal in $\mathbf{Set}$) to internal categories in $\mathcal{S}$, so you could thereby apply this theorem to the former.