Corollary 4.8.3.5. Let $n$ be an integer and let $\operatorname{\mathcal{C}}$ be an $(n,1)$-category. Then, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is weakly $(n-1)$-coskeletal.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. For $n < 0$, the result is trivial (see Example 4.8.1.9). We will therefore assume that $n \geq 0$. It follows from Corollary 4.8.1.23 that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) = \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $ is also $(n,1)$-category; in particular, it is minimal in dimensions $\geq n$. Since $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $(n-1)$-truncated (Corollary 4.8.1.20), it is locally $(n-2)$-truncated (Example 4.8.2.4). Applying Proposition 4.8.3.1, we conclude that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is weakly $(n-1)$-coskeletal. $\square$