Corollary 3.3.6.8. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. Then $f$ is a weak homotopy equivalence if and only if $\operatorname{Ex}^{\infty }(f)$ is a weak homotopy equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{ \rho ^{\infty }_ X} \ar [r]^-{f} & Y \ar [d]^{ \rho ^{\infty }_{Y} } \\ \operatorname{Ex}^{\infty }(X) \ar [r]^-{ \operatorname{Ex}^{\infty }(f) } & \operatorname{Ex}^{\infty }(Y), } \]
where the vertical maps are weak homotopy equivalences (Proposition 3.3.6.7). The desired result now follows from the two-out-of-three property (Remark 3.1.6.16). $\square$