TITLE: Stack associated to Lie group and manifold
QUESTION [8 upvotes]: Given a Lie group $G$, we have a Lie groupoid $(G\rightrightarrows *)$  and stack $BG=B\mathcal{G}$  of principal $G$ bundles.
Given a smooth manifold $M$, we have Lie groupoid $(M\rightrightarrows M)$  and stack $\underline{M}$ whose objects are smooth  maps to $M$.
Given Lie group $G$, we have two Lie groupoids associated to it :

$(G\rightrightarrows *)$ if we consider Lie group structure.
$(G\rightrightarrows G)$ if we ignore group structure and treat it as a manifold.

We have corresponding stacks associated :

$(G\rightrightarrows *)$ gives stack $B(G\rightrightarrows *)$, usually denoted by $BG$.
$(G\rightrightarrows G)$ gives stack $B(G\rightrightarrows G)$, usually denoted by $\underline{G}$.

As any Lie group is a manifold, shouldn't there be some relation with notions $BG$ and $\underline{G}$? I see they are not same. 
How are they related?
It is not even the case that the Lie groupoid $(G\rightrightarrows *)$ is pull back of $(G\rightrightarrows G)$ or the otherway around.

I do not know counter part in Algebraic geometry. 
Feel free  to (I request you to) relate  this to algebraic geometry version of stacks.

REPLY [5 votes]: $\underline{G}$ 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.