Kerodon

Remark 4.8.7.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. It follows from Remark 4.8.7.5 that if $F$ is categorically $(n+1)$-connective, then the induced map of homotopy $n$-categories $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})} \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{D}})}$ is an equivalence. In particular, if $F$ is categorically $2$-connective, then it induces an equivalence of homotopy categories $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$.