Warning 9.2.7.12. In the situation of Proposition 9.2.7.11, the comparison map $\theta $ need not be an isomorphism of simplicial sets. However, it is always bijective on $0$-simplices: vertices of both $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{A/ \, /Y} }( \widetilde{B}, \widetilde{X})$ and $\operatorname{Sol}( \sigma )$ can be identified with solutions to the lifting problem $\sigma $.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$