Warning 9.2.4.4. If $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, then the morphism $f: C \rightarrow C'$ of Theorem 9.2.4.3 can be chosen to depend functorially on $C$. Beware that this is generally not possible if $\operatorname{\mathcal{C}}$ is an $\infty $-category (see Example 9.2.4.5).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$