Given a group $G$ with delooping $BG$, $BG$ has a unique point $\iota:\star\rightarrow BG$ and $G$ is equivalent to the 2-pullack of $\iota$ along itself. Furthermore, the repeated 2-pullback $\iota\times_{BG} \iota \times_{BG} \iota$ is equivalent to $G\times G$ and comes with two morphisms $\tau_1,\tau_2:G\times G\rightarrow G$ such that, up to equivalence, $\tau_1$ can be identified with a projection and $\tau_2$ with the group multiplication, and this part is making me problems. I know this is common folklore and has already been vastly generalized, but I couldn't find a proof of it, particularly in the setting of (weak) 2-categories. I get the intuition, that, for a tuple $(g,h)$, if $\tau_1(g,h)= g$ then $\tau_2(g,h)$ has to be related to $g$ by composition with $h$ in $BG$, but I feel like I really need to stare at a proof of it to get familiar with the technical details.
Does anyone have a reference or could write down the proof?