G is the homotopy loop space of BG.

More precisely, the two terminal maps G→pt and G→pt yield a weak equivalence G → pt ⨯_{BG} pt, where the right side denotes the homotopy pullback of the diagram pt→BG←pt and pt denotes the representable stack of a smooth manifold given by a single point.

$\underline{G}$ is the homotopy loop space of $BG$.

More precisely, the two terminal maps $G\rightarrow pt$ and $G\rightarrow pt$ yield a weak equivalence $\underline{G} \rightarrow pt\times_{BG} pt$, where the right side denotes the homotopy pullback of the diagram $pt\rightarrow BG\leftarrow pt$ and $pt$ denotes the representable stack of a smooth manifold given by a single point.