Proposition 9.5.6.15. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$ if and only if it is accessibly embedded and closed under small limits in $\operatorname{\mathcal{C}}$.
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Proof. Assume that the subcategory $\operatorname{\mathcal{C}}_0$ is accessibly embedded and closed under small limits; we will show that $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$ (the converse is a special case of Variant 7.1.4.31). Since the $\infty $-category $\operatorname{\mathcal{C}}_0$ is accessible and complete, it is presentable. By assumption, the inclusion functor $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is accessible and continuous. Applying the adjoint functor theorem (Theorem 9.5.2.1), we conclude that $\iota $ admits a left adjoint $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$, so that $\operatorname{\mathcal{C}}_0$ is a reflective subcategory of $\operatorname{\mathcal{C}}$ (Proposition 6.2.2.18). $\square$