$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.2.1.20. 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 7.2.1.19 with Proposition 7.2.1.6.
$\square$