Let $G$ be a semisimple algebraic group, $C$ be a smooth projective curve, and $\omega$ be canonical line bundle.

The stack $Higgs_{\omega}$ is defined as the stack associating to each $S$ the groupoid consisting of $(E, \phi)$, where $E$ is a $G$-torsor over $X \times S$ and $\phi \in \Gamma(C \times S, ad(E) \otimes_C D)$. Here, how given this torsor $E$ on $C \times S$, does $ad(E)$ refers to the associated bundle $E \times_G \mathfrak{g}$?

**Main question:** Given a stack $X$, one can abstractly define its co-tangent stack. How does one show that the abstract definition of $T^* Bun_G$ can be identified with $Higgs_{\omega}$?