Let $G$ be a (discrete) group, and $1/G$ the corresponding groupoid with one object.  Consider the diagram in (the 2-category) Groupoids with one vertex, labeled $1/G$, the one arrow from that vertex to itself, given by the identity map.  
$$ \begin{matrix} 1/G \\ {\huge \circlearrowleft} \\ \scriptstyle \mathrm{id} \end{matrix}$$
(This diagram is equivalent to the pair of parallel arrows $1/G \overset{\rm id}{\underset{\rm id}\rightrightarrows} 1/G$.  Note that I am _not_ filling in the loop with a 2-cell.)

A cute fact is that the ("2-") limit of this diagram in Groupoids is the action groupoid $G/G$ of the adjoint action of $G$ on itself.  (See e.g. [2 limit in nLab](http://ncatlab.org/nlab/show/2-limit) or [HTT Chapter 4](http://arxiv.org/abs/math/0608040v4) for a definition of limits.)

Now, in homotopological terms, the groupoid $1/G$ looks like the classifying space ${\rm B}G$, and the above diagram looks like ${\rm B}G \times S^1$.  I have the possibly-mistaken impression that limits are supposed to look like topological cones (but maybe this is because we use words like "cone" when talking about limits).

__Question:__ In terms of homotopy, how should I visualize the limit cone
$$ \lim\left( \begin{matrix} 1/G \\ {\huge \circlearrowleft} \\ \scriptstyle \mathrm{id} \end{matrix} \right) $$
?