**Problem summary:** I'm trying to get some intuition for what the moduli space of objects for a dg-category (as in this paper by Brav and Dyckerhoff) actually looks like/how to give an alternative identification of the moduli of objects in certain cases.

**Edited to add:** Adding as per David's suggestion below: the original paper on the topic by Toën and Vaquié, and this paper on a coherent version of the construction by Lowrey that I was previously unaware of.

Let $\mathcal{C}$ be a compactly generated dg-category. Then the moduli of objects is defined such that on an affine $U$, $\mathcal{M}_{\mathcal{C}}(U)$ is the space of exact functors from compact objects $\mathcal{C}^c$ in $\mathcal{C}$ to perfect complexes on $U$. When $\mathcal{C}$ is smooth, such functors are corepresented by *left proper* objects $c\in \mathcal{C}^c\otimes\text{Perf}(U)$ -- this means that inner hom out of $c$ is a continuous functor $\mathcal{C}^c\otimes\text{Perf}(U) \to \text{Perf}(U)$ with a continuous right adjoint.

I'd like to get a better grasp on what these left proper objects actually look like. For instance, if $\mathcal{C} = \text{QCoh}(T^\ast C)$ for $C$ a smooth projective curve, then $\mathcal{M}_\mathcal{C}$ should be (I think) the moduli space of Higgs sheaves on $C$. If $\mathcal{C} = k[x]\text{-mod}$, then I think that being left proper should be related to some sort of ''finite support'' condition. But I'm not sure how to actually *prove* either of these.

To be concrete, let's suppose that $A$ is a regular Noetherian $k$-algebra for $k$ a field of characteristic zero, and consider th dg-category $A\text{-mod}$. If I have everything straight in my head (already a bold assumption), then I think $\text{Hom}_A(M,-): A\text{-mod}\to k\text{-mod}$ is continuous with a right adjoint precisely if $M$ is a perfect complex. I also think that in this case the right adjoint is the functor $\text{Hom}_k(M^\vee, -)$ where $M^\vee = \text{Hom}_A(M,A)$. If this is all the case, then what additional constraint is imposed on $M$ by continuity of its right adjoint?