Kerodon

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

Example 1.4.1.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be ordinary categories. It follows from Proposition 1.2.2.1 that the formation of nerves induces a bijection

\[ \xymatrix { \{ \text{Functors of ordinary categories from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$} \} \ar [d]^{\sim } \\ \{ \text{Functors of $\infty $-categories from $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$} \} . } \]

In other words, Definition 1.4.0.1 can be regarded as a generalization of the usual notion of functor to the setting of $\infty $-categories.