Kerodon

Proof of Proposition 9.5.7.10. Let $\overline{W}$ be the collection of morphisms $w$ of $\operatorname{\mathcal{C}}$ such that $H(w)$ is an isomorphism in $\operatorname{\mathcal{E}}$. As in the proof of Proposition 9.5.7.5, we see that $\overline{W}$ is the collection of objects of a presentable full subcategory $\operatorname{\mathcal{W}}\subseteq \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$, and is therefore generated under small colimits by a small subset $W \subseteq \overline{W}$. Let $\operatorname{\mathcal{D}}\subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $W$-local objects. It follows from Theorem 9.5.7.1 that $\operatorname{\mathcal{D}}$ is a Bousfield localization of $\operatorname{\mathcal{C}}$, so the inclusion functor $\operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}$ admits a left adjoint $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Applying Proposition 9.5.7.2, we see that there is an essentially unique cocontinuous functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ such that $G \circ F$ is isomorphic to $H$. Concretely, we can identify $G$ with the restriction $H|_{\operatorname{\mathcal{D}}}$. To complete the proof, it will suffice to show that $G$ is conservative. In other words, we wish to show that if $w$ is a morphism of $\operatorname{\mathcal{D}}$ which belongs to $\overline{W}$, then $w$ is an isomorphism. This is a special case of Remark 6.2.3.8, since every object of $\operatorname{\mathcal{D}}$ is $\overline{W}$-local. $\square$