Proposition 7.2.8.5. Let $f: K \rightarrow K'$ be a right cofinal morphism of simplicial sets. If $K$ is sifted, then $K'$ is also sifted.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Fix a finite set $I$. We have a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ K \ar [r]^-{\delta _{K}} \ar [d]^{f} & K^{I} \ar [d]^{f^{I}} \\ K' \ar [r]^-{\delta _{K'}} & K'^{I}, } \]
where the vertical maps are right cofinal (Corollary 7.2.1.20). Our assumption that $K$ is sifted guarantees that $\delta _{K}$ is right cofinal, so that $\delta _{K'}$ is also right cofinal (Proposition 7.2.1.6). $\square$