Kerodon

Remark 4.8.6.10 (Homotopy Invariance). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories and let $n$ be an integer. If $F$ is an equivalence of $\infty $-categories, then $G \circ F$ is essentially $n$-categorical if and only if $G$ is essentially $n$-categorical. If $G$ is an equivalence of $\infty $-categories, then $G \circ F$ is essentially $n$-categorical if and only if $F$ is essentially $n$-categorical. Both assertions are special cases of Remark 4.8.6.7.