Example 7.7.2.22. Let $\operatorname{\mathcal{C}}$ be (the nerve of) the category of sets, and let $K = B_{\bullet }G$ be the classifying simplicial set of a group $G$ (see Construction 1.3.2.5). Then a diagram $F: K \rightarrow \operatorname{\mathcal{C}}$ can be identified with a set $X$ equipped with an action of the group $G$, and the colimit $\varinjlim (F)$ can be identified with the quotient set $X/G$. This colimit is always universal (Example 7.7.0.3). Applying Remark 7.7.2.21, we see that the colimit of $F$ is strongly universal if and only if the following condition is satisfied:
- $(\ast )$
For every set $X'$ with an action of $G$ and every $G$-equivariant map $f: X' \rightarrow X$, the diagram of sets
\[ \xymatrix { X' \ar [r] \ar [d]^{f} & X'/G \ar [d] \\ X \ar [r] & X/G } \]is a pullback square.
Beware that this condition is not automatic: it is satisfied if and only if $G$ acts freely on $X$.