Proposition 6.37.
The forgetful functor induces an equivalence of categories
\[\Bun_{\Gg}(B) \iso \Act_{\Gg}(T) \times_{T} \{B\}.\]
Proof
We construct an explicit inverse. Consider a \(\Gg\)-action \(c\colon\Gg \ltimes P\to\Gg\) in \(T\) equipped with an isomorphism \(B \iso P\quot\Gg\). The colimit cocone exhibiting \(B\) as the realization of \(\Gg\ltimes P\) supplies compatible maps \((\Gg\ltimes P)_n\to B\) and hence lifts the action groupoid to a groupoid object in \(T_{/B}\). Pairing these structure maps with \(c\) gives a morphism of groupoid objects
\[\Gg\ltimes P\longrightarrow\Gg\times B\]
in \(T_{/B}\). Its underlying morphism in \(T\) is \(c\), so it is cartesian and therefore defines a \((\Gg\times B)\)-action in the slice. This construction defines a functor \[\Act_{\Gg}(T) \times_{T} \{B\} \to \Act_{\Gg\times B}(T_{/B}).\]
The map \(P\to B\) is an effective epimorphism because it is the atlas of the realization of the groupoid object \(\Gg\ltimes P\), and its quotient in \(T_{/B}\) is the terminal object \(B\to B\). Thus Lemma 6.36 shows that the resulting action is principal. Conversely, starting from a principal bundle, the same lemma identifies its quotient with \(B\), and the colimit cocone recovers its original structure maps to \(B\). Hence the two constructions are inverse.