$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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