Let $\mathcal{A}_g$ be the moduli space (stack?) of $g$-dimensional principally polarized abelian varieties.
We have the universal family of abelian varieties $\chi_g\rightarrow \mathcal{A}_g$, where the fiber over any $(A,\theta)$ is the corresponding principally polarized abelian variety. Let $L$ be the universal line bundle corresponding to the universal theta divisor.
If $(A,\theta)\in\mathcal{A}_g$. Let $V$ be a rank $r$ vector bundle on $A$ with chern classes $c_i$.
1) I was told that there is a family of coarse moduli spaces $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ parametrizing basically pairs $(A,F)$ where $A\in \mathcal{A}_g$ and $F$ is a coherent $\mu_{L|_{A}}$ -semistable sheaf on $A$ with rank $r$ and chern classes $c_i$. That is $F\in\mathcal{M}_A(r,c_i)$. Are there any conditions for this space to exist?
2) Consider the fiber product of $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)\times_{\mathcal{A}_{g}}\chi_g$. Can we find a universal sheaf (or a quasi-universal sheaf) $\mathcal{E}$ on this product, which when restricted to the fiber over any $(A,F)\in\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ gives $F$? When can we find such a sheaf?
How do I make these notions precise?
It would be great if someone can also direct me to references of the above especially of (1) and (2). Thanks in advance!
Edits: I have made edits based on Prof. Jason Starr's comment
