Kerodon

Remark 9.3.1.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories. Then an object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated if and only if the image $Y = F(X)$ is an $n$-truncated object of $\operatorname{\mathcal{D}}$. The “if” direction follows from Remark 9.3.1.6. For the converse, suppose that $X$ is $n$-truncated and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a homotopy inverse to $F$. Then $G(Y) \in \operatorname{\mathcal{C}}$ is isomorphic to $X$, and is therefore an $n$-truncated object of $\operatorname{\mathcal{C}}$ (Remark 9.3.1.5). Since $G$ is fully faithful, Remark 9.3.1.6 guarantees that $Y$ is an $n$-truncated object of $\operatorname{\mathcal{D}}$.