# Kerodon

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Corollary 7.1.3.22 (Limits in a Reflective Localization). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a reflective subcategory (Definition 6.2.2.1). Then a diagram $u: K \rightarrow \operatorname{\mathcal{C}}_0$ admits a limit in $\operatorname{\mathcal{C}}_0$ 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 $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$.

Proof. By virtue of Proposition 6.2.2.8, the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint. Invoking Corollary 7.1.3.21, we deduce that if $u$ admits a limit in $\operatorname{\mathcal{C}}_0$, then that limit is preserved by the inclusion functor (and therefore $u$ admits a limit in $\operatorname{\mathcal{C}}$). For the converse, assume that $u$ admits a limit $C$ in the $\infty$-category $\operatorname{\mathcal{C}}$. We will complete the proof by showing that $C$ is isomorphic to an object $C' \in \operatorname{\mathcal{C}}_0$; in this case, $C'$ is also a limit of $u$ in the $\infty$-category $\operatorname{\mathcal{C}}$ (Proposition 7.1.1.12), and therefore also a limit of $u$ in the $\infty$-category $\operatorname{\mathcal{C}}_0$ (Remark 7.1.1.10). Replacing $\operatorname{\mathcal{C}}_0$ by its essential image, we may assume that $\operatorname{\mathcal{C}}_0$ is replete. It follows that $\operatorname{\mathcal{C}}_0$ is the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $W$-local objects, for some collection of morphisms $W$ of $\operatorname{\mathcal{C}}$ (Proposition 6.3.3.9). We are therefore reduced to showing that the object $C = \varprojlim (u)$ is $W$-local, which follows from Proposition 7.1.1.14. $\square$