Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.5.5.9. Let $f: X \rightarrow X'$ and $g: Y \rightarrow Y'$ be categorical equivalences of simplicial sets. Then the induced map $(f \star g): X \star Y \rightarrow X' \star Y'$ is also a categorical equivalence of simplicial sets.

Proof. We have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \diamond Y \ar [r]^-{f \diamond g} \ar [d]^{ c_{X,Y} } & X' \diamond Y' \ar [d]^{ c_{X',Y'} } \\ X \star Y \ar [r]^-{ f \star g} & X' \star Y',} \]

where $f \diamond g$, $c_{X,Y}$, and $c_{X',Y'}$ are categorical equivalences (Remark 4.5.5.7 and Theorem 4.5.5.8). Invoking the two-out-of-three property (Remark 4.5.2.5), we conclude that $f \star g$ is also a categorical equivalence. $\square$