Variant 3.4.5.7. Let $X$ be a simplicial set, and let $\iota : X \hookrightarrow X^{+}$ be the inclusion map. Then the map $\operatorname{Ex}(\iota ): \operatorname{Ex}(X) \hookrightarrow \operatorname{Ex}(X^{+} )$ is a weak homotopy equivalence of semisimplicial sets.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Proposition 3.4.5.4, the counit map $v_{X}: X^{+} \rightarrow X$ is a weak homotopy equivalence of simplicial sets. Applying Corollary 3.3.5.2, we deduce that the map $\operatorname{Ex}(v_{X}): \operatorname{Ex}( X^{+} ) \rightarrow \operatorname{Ex}(X)$ is a weak homotopy equivalence of simplicial sets, hence also a weak homotopy equivalence of the underlying semisimplicial sets (Corollary 3.4.5.5). Since the composite map
\[ \operatorname{Ex}(X) \xrightarrow { \operatorname{Ex}(\iota ) } \operatorname{Ex}(X^{+} ) \xrightarrow { \operatorname{Ex}(v_{X} )} \operatorname{Ex}(X) \]
is the identity, it follows that $\operatorname{Ex}(\iota )$ is also a weak homotopy equivalence of semisimplicial sets. $\square$