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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 $[\mathbf C^{\rm op},\, \mathcal{S}] \cong [(\mathbf C \times \mathbf D)^{\rm op},\, \mathcal{Sets}]$ 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 $[\mathbf C^{\rm op},\, \mathcal S]$ of contravariant functors from $\bf C$ to $\mathcal S$ a topos?

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    $\begingroup$ If $\mathcal{S}$ is a Grothendieck topos, or more generally has large enough disjoint universal coproduct you are always fine. But If $\mathcal{S}$ is an elementary topos in general it is not going to work, but I do not know if there are nice conditions under which it works. $\endgroup$ – Simon Henry Jan 16 '20 at 23:40
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    $\begingroup$ If $\mathbf C$ were an internal category in $\mathcal S$, then the category of internal $\mathcal S$-valued presheaves on $\mathbf C$ would be a topos. (This is surely in Johnstone's "Topos Theory".) I think the point of @SimonHenry's comment is that good coproducts let you regard any genuine small category as an internal category in $\mathcal S$. $\endgroup$ – Andreas Blass Jan 17 '20 at 0:09
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    $\begingroup$ That was indeed where I was going. But the question seem much more interesting that this observation: For example if the category C is a groupoid I think that S-valued presheaf will be an elementary topos without needing any assumption on S, so the general question definitely do not reduce to the case I was refering too. $\endgroup$ – Simon Henry Jan 17 '20 at 0:16
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    $\begingroup$ The category of $G$-sets for a large group $G$ is a cocomplete elementary topos but is not a Grothendieck topos. [1] $\endgroup$ – Zhen Lin Jul 20 '20 at 3:33
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    $\begingroup$ @ZhenLin good to see you again! Yes, that's definitely relevant, and that example generalises a lot to other special large limits of toposes. $\endgroup$ – theHigherGeometer Jul 20 '20 at 6:01
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Rather than a full-fledged answer, this is a sketch of a plan:

As you yourself pointed out, if $\mathcal S=[\bf D^{\rm op},\, \mathcal \cal{Sets}] $, the result is trivial.

Now, suppose you consider an arbitrary elementary topos $\mathcal S$. The steps would be:

  1. Use the representation theorem of Joyal-Tierney to embed your target topos as the equivariant sheaves over a localic topos, see here.

    $E:\mathcal S \to Set^{\mathcal{L}^{op}}$

  2. Now each map $[\bf C^{\rm op},\, \mathcal S]$ composes with the embedding $E$ and thus lands into a localic topos.

  3. The last step would be to see how $[\bf C^{\rm op},\, E(\mathcal S)]$ sits inside $[\bf C^{\rm op},\, Set^{\mathcal{L}^{op}}]$.

Conjecture: under some some conditions (to be determined, but see this post When is a reflective subcategory of a topos a topos? ) there is a reflection which is sufficiently exact to ensure that it is indeed a subtopos of $[(\bf C \times \bf\mathcal{L})^{\rm op},\, \cal{Sets}]$

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