Let $P_{univ}$ denote this universal bundle. I believe the correct definition of $P_{univ}$ is this: It is the fibered category whose objects over a scheme $S$ are pairs $(P,s)$ where $P$ is a principal $G$-bundle over $C\times S$, and $s$ is a section of $P$, and morphisms are morphisms of bundles preserving the section. The forgetful map to $M_{G,C}$ is visibly representable, and one checks that the pullback of $P_{univ}$ via any map $f : S\rightarrow M_{G,C}$ is precisely the bundle corresponding to $f$. This is probably what Sorger means.