Variant 7.1.4.25 (Limits in a Reflective Localization). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory. Then a diagram $u: K \rightarrow \operatorname{\mathcal{C}}'$ admits a limit in $\operatorname{\mathcal{C}}'$ if and only if it admits a limit in $\operatorname{\mathcal{C}}$. In this case, the limit of $u$ is preserved by the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Variant 7.1.4.25. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory, and let $u: K \rightarrow \operatorname{\mathcal{C}}'$ be a diagram. Applying Corollary 6.2.2.12, we deduce that $\operatorname{\mathcal{C}}'_{/u}$ is a reflective subcategory of $\operatorname{\mathcal{C}}_{/u}$. If the diagram $u$ admits a limit in $\operatorname{\mathcal{C}}$, then the slice $\infty $-category $\operatorname{\mathcal{C}}_{/u}$ has a final object $X$. Applying Lemma 7.1.4.26, we deduce that $X$ is isomorphic to an object of $\operatorname{\mathcal{C}}'_{/u}$, which is also a final object of $\operatorname{\mathcal{C}}_{/u}$ and therefore also of $\operatorname{\mathcal{C}}'_{/u}$. In particular, the diagram $u$ has a limit in $\operatorname{\mathcal{C}}'$ (which is preserved by the inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$). $\square$