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

The stack $\mathrm{Higgs}_{\omega}$ 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 $\phi \in \Gamma(C \times S, \operatorname{ad}(E) \otimes_C D)$. Here, how, given this torsor $E$ on $C \times S$, does $\operatorname{ad}(E)$ refer 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 $T^* \mathrm{Bun}_G$ can be identified with $\mathrm{Higgs}_{\omega}$?