Remark 4.8.2.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories. If $\operatorname{\mathcal{D}}$ is locally $n$-truncated, then $\operatorname{\mathcal{C}}$ is locally $n$-truncated. The converse holds if $F$ is an equivalence of $\infty $-categories. In particular, if $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are equivalent, then $\operatorname{\mathcal{C}}$ is locally $n$-truncated if and only if $\operatorname{\mathcal{D}}$ is locally $n$-truncated.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$