Kerodon

Proof. Fix an object $C \in \operatorname{\mathcal{C}}$, and let $h_{C}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ denote the functor represented by $C$. We wish to show that, for every morphism $w: X \rightarrow Y$ which belongs to $W$, the image $h_{C}(w)$ is a homotopy equivalence of Kan complexes. By virtue of Remark 3.5.1.19, it will suffice to show that $h_{C}(w)$ is $n$-connective for every integer $n \geq 0$. The proof proceeds by induction on $n$. In the case $n = 0$, we wish to show that the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,C) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,C)$ is surjective on connected components, which follows from our assumption that $C$ is weakly $W$-local. Let us therefore assume that $n > 0$. Using the criterion of Corollary 3.5.1.29 (together with Exercise 7.6.4.13), we are reduced to proving the $(n-1)$-connectivity of the relative diagonal of $h_{C}(w)$ (formed in the $\infty $-category $\operatorname{\mathcal{S}}$). Since the functor $h_{C}$ preserves limits (Proposition 7.4.5.16), we can identify the relative diagonal of $h_{C}(w)$ with $h_{C}( \gamma _{X/Y} )$, where $\gamma _{X/Y}: Y \coprod _{X} Y \rightarrow Y$ denotes a relative codiagonal of $w$. By assumption, we can arrange that $\gamma _{X/Y}$ is also contained in $W$, so the desired result follows from our inductive hypothesis. $\square$