Corollary 4.4.3.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a replete subcategory. Then the Kan complex $\operatorname{\mathcal{C}}_{0}^{\simeq }$ is a summand of the Kan complex $\operatorname{\mathcal{C}}^{\simeq }$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. The assumption that $\operatorname{\mathcal{C}}_0$ is replete guarantees that the inclusion map $\iota : \operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is an isofibration (Example 4.4.1.12). Applying Proposition 4.4.3.7, we deduce that the inclusion $\operatorname{\mathcal{C}}_0^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}^{\simeq }$ is a Kan fibration, so that $\operatorname{\mathcal{C}}_0^{\simeq }$ is a summand of $\operatorname{\mathcal{C}}^{\simeq }$ by virtue of Example 3.1.1.4. $\square$