Definition 6.3.6.1. Let $f: X \rightarrow S$ be a morphism of simplicial sets. We will say that $f$ is *universally localizing* if, for every morphism of simplicial sets $S' \rightarrow S$, the projection map $S' \times _{S} X \rightarrow S'$ exhibits $S'$ as a localization of $S' \times _{S} X$ with respect to some collection of edges $W$.

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