Theorem 6.38. (Classification of principal bundles)
Let \(\Gg\) be a groupoid object in \(T\) and let \(B \in T\) be an object. There is an equivalence of categories
\[\Hom_{T}(B,\bB\Gg) \iso \Bun_{\Gg}(B),\]
given on objects by sending a morphism \(c\colon B \to \bB\Gg\) to the principal \(\Gg\)-bundle \(P \to B\) defined by the pullback square
Proof
Recall from Proposition 6.30 the equivalence \(-\quot\Gg \colon \Act_{\Gg}(T) \iso T_{/\bB\Gg}\), which induces an equivalence on fibers over \(T\):
\[\Act_{\Gg}(T) \times_{T} \{B\} \;\simeq\; T_{/\bB\Gg} \times_{T} \{B\} \;\simeq\; \Hom_{T}(B,\bB\Gg).\]
The inverse sends a morphism \(c\colon B \to \bB\Gg\) to the base change \(P = B \times_{\bB\Gg} \Gg_0 \twoheadrightarrow B\) of the atlas \(\Gg_0 \twoheadrightarrow \bB\Gg\) along \(c\). Combining with the equivalence from Proposition 6.37 finishes the proof.