Kerodon

Remark 4.8.6.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. For $n \geq 0$, $F$ is essentially $n$-categorical if and only if the diagonal map $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is essentially $(n-1)$-categorical. This follows by combining Remark 4.8.6.12 with Corollary 3.5.9.17, since $F$ induces a Kan fibration $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ (Proposition 4.6.1.21).