Proposition 4.8.2.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. If $\operatorname{\mathcal{C}}$ is an $n$-coskeletal simplicial set (Definition 3.5.3.1), then it is locally $(n-2)$-truncated.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, our assumption that $\operatorname{\mathcal{C}}$ is $n$-coskeletal guarantees that the pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ is $(n-1)$-coskeletal (Remark 4.6.5.4). In particular, it is $(n-2)$-truncated (Example 3.5.7.2). The desired result now follows from Remark 4.8.2.5. $\square$