Universal principal bundle on stack I am studying the notes of Sorger concerning the moduli problem of principal bundles over curve https://inis.iaea.org/collection/NCLCollectionStore/_Public/38/005/38005695.pdf and there is something I don't quite understand. He considers the moduli stack of principal $G$-bundles over a curve $G$, and it is denoted by $\mathcal{M}_{G,C}$, being $\mathcal{M}_{G,C}(S)$ the groupoid of principal $G$-bundles over $C\times S$ for each $\mathbb{C}$-scheme $S$. In Proposition 3.6.8, he considers "the universal principal $G$-bundle on $C\times\mathcal{M}_{G,C}$" and I don't have a clue about what this is. I know that giving a principal bundle over a stack is to give for each $\mathbb{C}$-scheme $S$ a principal bundle over $S$, for each morphism of schemes $S\rightarrow S'$ an isomorphism between the principal bundle over $S$ and the pullback of the principal bundle over $S'$ by the map $S\rightarrow S'$, and in such way that the cocycle compatibility condition holds.  I would really like to know what is the universal principal bundle in this case. Thank you for your time and wisdom in advance.
 A: Expanding on abx’s comment, let’s suppose $\mathcal{M}_{G,C}$ was representable as a scheme. Then for any $\mathbb{C}$-scheme $S$ there is a canonical isomorphism
$Hom_\mathbb{C}(S, \mathcal{M}_{G,C}) \cong \{\text{Groupoid of principal }G\text{-bundles over }C \times S\}$.
Now take $S = \mathcal{M}_{G,C}$, and consider the element of the LHS given by the identity map on $\mathcal{M}_{G,C}$. The corresponding element of the RHS is precisely the “the universal principal $G$-bundle on $C\times\mathcal{M}_{G,C}$". It is called this because any other principal $G$-bundle on some $C \times T$ can be uniquely realized as a pullback of this one via some morphism $T\rightarrow \mathcal{M}_{G,C}$.
Now in reality, $\mathcal{M}_{G,C}$ is not a scheme but merely an algebraic stack. However, there is still a notion of morphisms $T\rightarrow \mathcal{M}_{G,C}$ for $\mathbb{C}$-schemes $T$, and some stacky notion of bundle on it which pulls back to honest bundles on schemes. I can’t fill in the details but I hope this gives some intuition.
