Remark 9.3.1.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects $Y \in \operatorname{\mathcal{C}}$ for which the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$ is $n$-truncated. Since the representable functor
carries colimits in the $\infty $-category $\operatorname{\mathcal{C}}$ to limits in the $\infty $-category of spaces $\operatorname{\mathcal{S}}$, (Corollary 7.4.1.19), Remark 7.4.1.5 guarantees that the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is closed under the formation of colimits. Consequently, if $\operatorname{\mathcal{C}}$ is generated (under the formation of small colimits) by some full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$, then the object $X$ is $n$-truncated if and only if the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$ is $n$-truncated for each object $Y \in \operatorname{\mathcal{C}}_0$.