Is there a *relative* moduli stack of objects functor?

Question

Toen and Vaquie have constructed for any dg category $\mathcal{C}$ a stack $\mathcal{M}_\mathcal{C}$ parametrising objects in $\mathcal{C}$. Its definition is $$\mathcal{M}_\mathcal{C}(R)\ =\ \text{Maps}_{\text{dgCat}}(\text{Perf} R, \mathcal{C})$$ for $R$ a ring.

1. It is an Artin stack if $\mathcal{C}$ is of finite type.
2. The functor $\mathcal{M}_{(-)}$ is right adjoint to the functor taking a 1-stack $X$ to the dg category $\text{Perf}(X)\subseteq \text{QCoh}(X)$.

My question is: if $\mathcal{C}$ is a sheaf of categories* over a space $B$ (scheme, stack, prestack, ... ) can we define a space $$\mathcal{M}_{\mathcal{C}/B}\ \stackrel{?}{\to}\ B,$$ in what generality is it defined (i.e. for an arbitrary dg category $\mathcal{C}$ and prestack $B$?), and are the analogues of 1. and 2. true?

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*e.g. if $B$ is affine this is the same thing as a dg category plus an algebra action of $\text{QCoh}(B)$.

Answer 1

Let me give the relative construction. We'll say our geometric objects are presheaves on some category $Aff$, and we'll denote sheaves of categories by $2QCoh(-)$.

There's a functor

$$P(Aff)_{/B}^{op} \to 2QCoh(B)$$

$$(f: X \to B) \mapsto Perf(X)$$ sending a presheaf over $B$ to a $Perf(X)$ viewed as a sheaf of categories over $B$, via $f^*$. One shows that this preserves limits, and the right adjoint $M_{(-)/B}$ is a relative moduli of objects, as a presheaf over $B$.

As in the absolute case we have an explicit formula

$$M_{C/B}(Spec(R) \to B) \simeq Map_{2QCoh(B)}(Perf(R), C)$$

where in the first argument on the RHS, $Perf(R)$ is endowed with the structure of a sheaf of categories over $B$ via the map $Spec(R) \to B$. And of course $B= *$ case recovers the absolute case.

I expect arguments of Antieau-Gepner about representability of moduli generalize to this relative setting, though I do not have a precise statement at the moment.

Footnote: As you do in the question statement I'm sweeping some presentability issues under the rug here to give the idea, $2QCoh(-)$ needs to be constructed carefully, but sounds like you know where to look for these constructions.