Given a curve $C$ of genus $g$ I know how to construct a quasi-projective variety $M_C(r,d)$ as the GIT quotient of a certain Quot scheme $Q=Quot^{r,d}(\mathcal{O}_C(-n)^N)$ by the action of a certain group, let's call it $G$, whose geometric points are the semi-stable vector bundles on $C$ of rank $r$ and degree $d$. This shows that $M_C(r,d)$ is quite close to being a coarse moduli space for the functor $$F(S)=\{\text{vector bundles } V \text{ on } C\times_kS \text{ of rank r and degree d} \}$$
(where by that we mean looking at the fibers $V(s)$ for $s \in S$). But how can one construct an actual natural transformation from $F$ to the functor $h^{M_C}$? It would do it if vector bundles in $F(S)$ could be written as quotients in a canonical way but I don't think this is possible. In particular I don't think one can use Mumford's theorem giving a polynomial bound on Castelnuovo regularity because it needs an embedding into a trivial vector bundle which I don't think we have.
Le Potier does as follows. He says that given a vector bundle $V$ then we have that $p_\ast V$ is locally free of rank its Euler characteristic $\chi$ (because 1/ we can choose $d$ as large as we want 2/ Grauert's lemma and Riemann-Roch (already one needs assumptions on $S$ but nevermind that)). He puts $$R=Isom(\mathcal{O}_S^\chi,p_\ast V)$$ as a principal $GL(\chi)$- bundle on $S$ (or so I think as he does not really explain), and then he says there's a $G$-equivariant map from $R$ to the Quot scheme $Q$ (which then gives a map $S=R/G \longrightarrow Q/G=M_C(r,d)$ as we want. However I have no idea what this map is. Could someone explain this to me, or show me another natural transformation making $M_C$ a coarse moduli space of $F$?