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.

- It is an Artin stack if $\mathcal{C}$ is of finite type.
- 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?

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