Remark 4.3.6.2. In the special case where $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, Proposition 4.3.6.1 follows from Corollary 4.3.5.17; in fact, both of the simplicial sets $\operatorname{\mathcal{C}}_{f/}$ and $\operatorname{\mathcal{C}}_{/f}$ are (the nerves of) ordinary categories.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$