$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 4.8.8.23. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. For every integer $n$, the following conditions are equivalent:
- $(1)$
The functor $F$ is essentially $n$-categorical.
- $(2)$
The functor $F$ factors as a composition $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{D}}' \xrightarrow {G} \operatorname{\mathcal{D}}$, where $F'$ is an equivalence of $\infty $-categories and $G$ is an $n$-categorical isofibration.
Proof.
The implication $(2) \Rightarrow (1)$ follows from Proposition 4.8.6.35 (together with Remark 4.8.5.18). To prove the converse, we may assume without loss of generality that $F$ is an isofibration (Corollary 4.5.2.23). In this case, the factorization $\operatorname{\mathcal{C}}\xrightarrow {F'} \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \xrightarrow {G} \operatorname{\mathcal{D}}$ of Remark 4.8.8.15 has the desired properties: Proposition 4.8.8.22 guarantees that $F'$ is an equivalence of $\infty $-categories, Proposition 4.8.8.21 guarantees that $G$ is an isofibration, and Proposition 4.8.8.14 guarantees that $G$ is $n$-categorical.
$\square$