Remark 5.3.7.9. In the situation of Corollary 5.3.7.7, it is not necessary to assume that $\operatorname{\mathcal{D}}$ is an $\infty $-category: every morphism of simplicial sets $f: X \rightarrow Z$ admits a factorization $X \xrightarrow {f'} Y \xrightarrow {f''} Z$, where $f''$ is an isofibration and $f'$ both a monomorphism and a categorical equivalence (Proposition ). However, the proof is somewhat more difficult.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$