Proposition 9.3.3.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n \geq -1$ be an integer, and let $X$ be an object of $\operatorname{\mathcal{C}}$ for which there exists a product $X \times X$. Then $X$ is $n$-truncated if and only if the diagonal map $\delta _{X}: X \rightarrow X \times X$ is $(n-1)$-truncated.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. For each object $C \in \operatorname{\mathcal{C}}$, Example 3.5.9.18 shows that the mapping space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X)$ is $n$-truncated if and only if the diagonal map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \]
is $(n-1)$-truncated. The desired result now follows by allowing the object $C$ to vary. $\square$