$\infty$-topos as an internal $\infty$-category in itself

Question

I'm interested (both autonomously and directly related to my work) in the natural internalization of $\infty$-topos sheaves in it (as usual, assuming Grothendieck universes). Is there any literature where this has arisen before?

I was especially interested in the following natural construction. Let C be an $\infty$-site and $H$ be the topos of $\infty$-sheaves on it. We call an object $A \in H$ $u$-small (for a Grothendieck universe $u$) if all its $\infty$-groupoids are realized by $u$-small Kan complexes. Now the subcategory $uH$ of all $u$-small objects of $H$ and $uH$ is an $\infty$-topos in the $u$-sense. Let us now define a (presumably) internal category $u\mathcal{H}$ in $H$ as

$(u\mathcal{H})_n(c) := \mathrm{core}((uH)^{[n]} \to c)$ (core of over-category)

and pullbacks along morphisms. This is actually a well-defined simplicial $\infty$-presheaf on $C$, right? (as far as I understand, in a $n$-topos we would get a presheaf of $(n+1)$-groupoids since non-trivial isomorphisms would arise between the pullbacks, but the perfection of the $\infty$-approach is that $\infty + 1 = \infty$). Under what conditions can we expect that this simplicial presheaf will actually be an internal category?

Answer 1

In Martini and Wolf's series on internal higher category theory, they prove an equivalence between sheaves of categories on an $\infty$-topos $B$ and internal categories in $B$.

The former is better suited to talk about large internal $B$-categories. From this perspective, your $u\mathcal H$ is the sheaf $b\mapsto B_{/b}$ with some cardinal restriction, but you might as well take the whole thing.

They call this object $\Omega_B$ or $\Omega$ and as you may expect it plays a central role in their theory (It is not to be mistaken with the $\Omega$ from classical topos theory which is often used denote the subobject classifier : this is more like an object classifier)