Corollary 8.1.2.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets. Then $F$ is a weak homotopy equivalence if and only if the induced map $\operatorname{Tw}(F): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{D}})$ is a weak homotopy equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We have a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{ \operatorname{Tw}(F) } \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{D}}, } \]
where the vertical maps are weak homotopy equivalences by virtue of Corollary 8.1.2.4. $\square$