Kerodon

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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.