Kerodon

Proof. Suppose that $f: X \rightarrow S$ is a morphism of simplicial sets which can be realized as the colimit of a filtered diagram $\{ f_{\alpha }: X_{\alpha } \rightarrow S_{\alpha } \} $ in the category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$, where each $f_{\alpha }$ is universally localizing. We wish to show that $f$ is universally localizing. Fix a morphism of simplicial sets $T \rightarrow S$ and let $W$ be the collection of all edges $w = (w_ T, w_ X)$ of $T \times _{S} X$ for which $w_{T}$ is a degenerate edge of $T$. Note that the projection map $f_{T}: T \times _{S} X \rightarrow T$ can be realized as a filtered colimit of morphisms $f_{T,\alpha }: T \times _{S} X_{\alpha } \rightarrow T \times _{S} S_{\alpha }$. For each index $\alpha $, let $W_{\alpha }$ denote the collection of edges of $T \times _{S} X_{\alpha }$ having degenerate image in $T$. Since $f_{\alpha }$ is universally localizing, Proposition 6.3.6.3 guarantees that $f_{T,\alpha }$ exhibits $T \times _{S} S_{\alpha }$ as a localization of $T \times _{S} X_{\alpha }$ with respect to $W_{\alpha }$. Applying Proposition 6.3.4.1, we conclude that $f_{T}$ exhibits $T$ as a localization of $T \times _{S} X$ with respect to $W$. $\square$