From my general experience I think for myself of what follows as some kind of taboo question for some reason: in my imagination, everybody wants an answer to this but somehow thinks it shall not be asked.

OK among many ways to present a stack, I choose this one: we are given a Grothendieck topos $\mathbf X$ represented by sheaves of sets on a site $(\mathbb C,J)$, and then we have "a large category $\mathscr C$ in the world of $\mathbf X$", that is, a presheaf $\mathbb C^{\mathrm{op}}\to\text{Categories}$ satisfying the (three-step) glueing conditions w. r. t. $J$. In this question, by stacks are meant such $\mathscr C$'s. More precisely, we speak about stacks on $\mathbf X$, or stacks on $(\mathbb C,J)$.

Given that, we may consider the notion of "presheaf of $\mathbf X$-world sets on $\mathscr C$ in the $\mathbf X$-world". Again, there are several ways to define this, for example, as another gadget $\mathscr E$ of the same kind as $\mathscr C$ together with a "functor in the $\mathbf X$-world" $\mathscr E\to\mathscr C$ which is a discrete fibration.

All such discrete fibrations form a category which I will denote by $\operatorname{Sets}(\mathbf X)^{\mathscr C^{\mathrm{op}}}$, since, I believe, it can be appropriately described as the category of contravariant functors, in the $\mathbf X$-world, from $\mathscr C$ to $\operatorname{Sets}(\mathbf X)$, the latter being yet another gadget of the same kind as $\mathscr C$, with the "underlying" $\mathbb C^{\mathrm{op}}\to\text{Categories}$ sending $c\in\mathbb C$ to $\mathbf X/a(h_c)$ (slice over the associated sheaf of $h_c:=\hom_{\mathbb C}(-,c)$).

The question now is simply this: under what conditions does it happen that there is another Grothendieck topos $\mathbf Y$ such that the category $\operatorname{Sets}(\mathbf X)^{\mathscr C^{\mathrm{op}}}$ is equivalent to $\mathbf Y$?

**Remarks**

I am primarily interested in the case when $\mathscr C$ is the associated stack of an internal category of $\mathbf X$. I believe in this case several things simplify.

Since $\mathscr C$ is in general not small (i. e. not the externalization, in a known way, of an internal category of $\mathbf X$), there is in general no well-defined geometric morphism $\operatorname{Sets}(\mathbf X)^{\mathscr C^{\mathrm{op}}}\to\mathbf X$, but even if there is no such morphism, I believe it is still natural to call $\mathbf Y$, when it exists, the total space of the stack $\mathscr C$.

Whereas if there is such a geometric morphism, it still might be different from the one with inverse image "constant presheaf" and direct image "$\varprojlim_{\mathscr C}$". Or it does coincide but is not bounded. Or further, although not bounded, is $\textit{locally}$ bounded. Hence subquestion: can such things happen?

There is a variation which might be needed to have more natural examples - when $\mathscr C$ comes naturally equipped with its own "$\mathbf X$-world Grothendieck topology" which one cannot ignore, i. e. one has to consider the $\mathbf X$-world $\textit{sheaves}$ rather than $\operatorname{Sets}(\mathbf X)^{\mathscr C^{\mathrm{op}}}$ to obtain something sensible.

Finally, the natural reverse questions are - which geometric morphisms $f:\mathbf Y\to\mathbf X$ are of this form? For those which are - what, if any, additional data on $f$ enable to recover the stack $\mathscr C$?