Corollary 9.5.6.20. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be presentable $\infty $-categories. Suppose we are given continuous accessible functors $G,G': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\alpha : G \rightarrow G'$. Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by those objects $C \in \operatorname{\mathcal{C}}$ such that $\alpha _{C}: G(C) \rightarrow G'(C)$ is an isomorphism in $\operatorname{\mathcal{D}}$. Then $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$.
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Proof. The natural transformation $\alpha $ can be identified with a continuous accessible functor $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})$. By construction, $\operatorname{\mathcal{C}}_0$ is the inverse image under $T$ of the full subcategory $\operatorname{Isom}(\operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})$ spanned by the isomorphisms. Since $\operatorname{Isom}( \operatorname{\mathcal{D}})$ is a Bousfield localization of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})$ (Example 9.5.6.16), the desired result is a special case of Corollary 9.5.6.19. $\square$