Corollary 9.3.3.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n \geq -1$ be an integer, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ for which there exists a fiber product $X \times _{Y} X$. Then $f$ is $n$-truncated if if and only if the relative diagonal $\delta _{X/Y}: X \rightarrow X \times _{Y} X$ is $(n-1)$-truncated (see Notation 7.6.2.15).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let us identify the morphism $f$ with an object $\overline{X}$ of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. By virtue of Proposition 7.6.2.14, there exists a product $\overline{X} \times \overline{X}$ in the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$, whose image in $\operatorname{\mathcal{C}}$ is the fiber product $X \times _{Y} X$. Moreover, the relative diagonal $\delta _{X/Y}$ can be identified with the image of the diagonal map $\delta _{ \overline{X} }: \overline{X} \rightarrow \overline{X} \times \overline{X}$ under the forgetful functor $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$. Applying Corollary 9.3.3.11, we see that $\delta _{X/Y}$ is an $(n-1)$-truncated morphism of $\operatorname{\mathcal{C}}$ if and only if $\delta _{ \overline{X}}$ is an $(n-1)$-truncated morphism of $\operatorname{\mathcal{C}}_{/Y}$. By virtue of Proposition 9.3.3.19, this is equivalent to the requirement that $\overline{X}$ is $n$-truncated as an object of $\operatorname{\mathcal{C}}_{/Y}$. The desired result now follows from the criterion of Proposition 9.3.3.7. $\square$