Proposition 7.2.8.6. Let $f: K \rightarrow K'$ be a categorical equivalence of simplicial sets. Then $K$ is sifted if and only if $K'$ is sifted.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. It will suffice to show that, for every finite set $I$, the diagonal map $\delta _{K}: K \rightarrow K^{I}$ is right cofinal if and only if the diagonal map $\delta _{K'}: K \rightarrow K'^{I}$ is right cofinal. This follows by applying Corollary 7.2.1.22 to the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ K \ar [r]^-{\delta _{K}} \ar [d]^{f} & K^{I} \ar [d]^{f^{I}} \\ K' \ar [r]^-{\delta _{K'}} & K'^{I}. } \]
$\square$