Variant 4.8.6.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. For $n \geq 0$, the functor $F$ is essentially $n$-categorical if and only if the composite map
\[ \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}}^{\mathrm{h}} \operatorname{\mathcal{C}} \]
is essentially $(n-1)$-categorical. To prove this, we can use Corollaries 4.5.2.23 and 4.5.2.20 to reduce to the situation where $F$ is an isofibration. In this case, the desired result is a reformulation of Remark 4.8.6.13 (see Corollary 4.5.2.28).