Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 4.4.1.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. Then $F$ is an isofibration (in the sense of Definition 4.4.1.1) if and only if the induced map of simplicial sets $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is an isofibration of $\infty $-categories. This follows from the observation that $\operatorname{N}_{\bullet }(F)$ is automatically an inner fibration (see Proposition 4.1.1.10).