Theorem 9.5.7.1. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category, let $W$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $W$-local objects. Then $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Corollary 9.5.6.21, we may assume that $W = \{ w\} $ consists of a single morphism $w: C \rightarrow D$ of $\operatorname{\mathcal{C}}$. Let $h^ C, h^{D}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ denote the functors corepresented by $C$ and $D$, respectively, so that $w$ induces a natural transformation $h^{w}: h^{D} \rightarrow h^{C}$. By definition, an object $X \in \operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}_0$ if and only if $h^{w}$ carries $X$ to an isomorphism in $\operatorname{\mathcal{S}}$. Since the functors $h^{C}$ and $h^{D}$ are continuous and accessible (Proposition 7.4.1.22 and Example 9.4.7.16), Corollary 9.5.6.20 guarantees that $\operatorname{\mathcal{C}}_0$ is a Bousfield localization of $\operatorname{\mathcal{C}}$. $\square$