Remark 9.2.6.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every collection of morphisms $S$ of $\operatorname{\mathcal{C}}$, let $[S]$ be the collection of homotopy classes of morphisms which belong to $S$. If $( S_{L}, S_{R} )$ is a weak factorization system on $\operatorname{\mathcal{C}}$, then $( [S_{L}], [S_{R} ] )$ is a weak factorization system on (the nerve of) the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. See Warning 9.2.5.5 and Variant 8.5.1.3.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$