$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.3.3.18 (Pullbacks of Truncated Morphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y' \ar [r] & Y } \]
and let $n$ be an integer. If $f$ is $n$-truncated, then $f'$ is also $n$-truncated.
Proof.
Let $C \in \operatorname{\mathcal{C}}$ be an object. Applying Proposition 7.4.1.18, we obtain a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X') \ar [r] \ar [d]^{ \theta '} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \ar [d]^{ \theta } \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y') \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,Y) } \]
in the $\infty $-category of spaces. Corollary 7.6.3.11 guarantees that if $\theta $ is $n$-truncated, then $\theta '$ is also $n$-truncated. Proposition 9.3.3.18 now follows by allowing the object $C$ to vary.
$\square$