Corollary 3.4.5.8. Let $X$ and $Y$ be simplicial sets and let $f: X \rightarrow Y$ be a morphism of semisimplicial sets. Then $f$ is a weak homotopy equivalence of semisimplicial sets if and only if the induced map $\operatorname{Ex}(f): \operatorname{Ex}(X) \rightarrow \operatorname{Ex}(Y)$ is a weak homotopy equivalence of semisimplicial sets.

**Proof.**
By definition, $f: X \rightarrow Y$ is a weak homotopy equivalence of semisimplicial sets if and only if the induced map $f^{+}: X^{+} \rightarrow Y^{+}$ is a weak homotopy equivalence of simplicial sets. By virtue of Corollary 3.3.5.2, this is equivalent to the assertion that $\operatorname{Ex}(f^{+}): \operatorname{Ex}(X^{+} ) \rightarrow \operatorname{Ex}(Y^{+} )$ is a weak homotopy equivalence when viewed as a morphism of simplicial sets, or equivalently when viewed as a morphism of semisimplicial sets (Corollary 3.4.5.5). The desired result now follows by inspecting the commutative diagram of semisimplicial sets

since the vertical maps are weak homotopy equivalences by virtue of Variant 3.4.5.7. $\square$