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Corollary Let $f: X \rightarrow Y$ and $f': X' \rightarrow Y'$ be left cofinal morphisms of simplicial sets. Then the product map $(f \times f'): X \times X' \rightarrow Y \times Y'$ is left cofinal.

Proof. Factoring $f \times f'$ as a composition

\[ X \times X' \xrightarrow { f \times \operatorname{id}_{X'} } Y \times X' \xrightarrow { \operatorname{id}_{Y} \times f'} Y \times Y', \]

the desired result follows by combining Corollary with Proposition $\square$