Corollary 4.7.2.9. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a right fibration of $\infty $-categories, let $n$ be an integer, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then $f$ is $n$-truncated if and only if $U(f)$ is an $n$-truncated morphism of $\operatorname{\mathcal{D}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Since $U$ is a right fibration, it induces a trivial Kan fibration $U_{/Y}: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{D}}_{ / U(Y) }$ (Corollary 4.3.7.13). Applying Remark 4.7.1.13, we deduce that an object of $\operatorname{\mathcal{C}}_{/Y}$ is $n$-truncated if and only if its image in $\operatorname{\mathcal{D}}_{ / U(Y) }$ is $n$-truncated. The desired result now follows from Proposition 4.7.2.7. $\square$