Corollary 9.4.8.14. Let $\operatorname{\mathcal{C}}$ be an accessible $\infty $-category, let $n$ be an integer, and let $\operatorname{\mathcal{C}}_{\leq n}$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $n$-truncated objects. Then $\operatorname{\mathcal{C}}_{\leq n}$ is an accessibly embedded full subcategory of $\operatorname{\mathcal{C}}$. In particular, $\operatorname{\mathcal{C}}_{\leq n}$ is accessible.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Fix a small collection of objects $\{ C_ i \} _{i \in I}$ which generates $\operatorname{\mathcal{C}}$ under colimits. For each $i \in I$, let $\operatorname{\mathcal{C}}_ i$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects $X$ for which the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_ i, X)$ is $n$-truncated. By construction, $\operatorname{\mathcal{C}}_ i$ is the inverse image of the full subcategory $\operatorname{\mathcal{S}}_{\leq n} \subseteq \operatorname{\mathcal{S}}$ under the fucntor
\[ h^{C_ i}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}\quad \quad X \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, X) \]
corepresented by $C_ i$. Since the functor $h^{C_ i}$ is accessible (Example 9.4.7.16) and the subcategory $\operatorname{\mathcal{S}}_{\leq n} \subseteq \operatorname{\mathcal{S}}$ is accessible embedded (Example 9.4.7.10), it follows that the subcategory $\operatorname{\mathcal{C}}_ i \subseteq \operatorname{\mathcal{C}}$ is also accessibly embedded. It follows from Remark 7.4.1.25 that $\operatorname{\mathcal{C}}_{\leq n}$ coincides with the intersection $\bigcap _{i \in I} \operatorname{\mathcal{C}}_ i$, and is therefore also an accessibly embedded subcategory of $\operatorname{\mathcal{C}}$ (Corollary 9.4.8.13). $\square$