Kerodon

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Corollary 6.3.6.11. Let $f: X \rightarrow S$ be a universally localizing morphism of simplicial sets, and let $K$ be a weakly contractible simplicial set. Then the composite map $X \times K \rightarrow X \xrightarrow {f} S$ is universally localizing.

Proof. By virtue of Proposition 6.3.6.10, it will suffice to show that the projection map $X \times K \rightarrow X$ is universally localizing. Using Remark 6.3.6.7, we can reduce to the case $X = \Delta ^0$, in which case the desired result follows from Remark 6.3.6.6. $\square$