Corollary 3.4.5.6. For every semisimplicial set $X$, the inclusion map $\iota : X \hookrightarrow X^{+}$ is a weak homotopy equivalence of semisimplicial sets.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We wish to show that the map $\iota ^{+}: X^{+} \rightarrow (X^{+})^{+}$ is a weak homotopy equivalence of simplicial sets. This is clear, since $\iota ^{+}$ is right inverse to the counit map $v_{ X^{+} }: (X^{+})^{+} \rightarrow X^{+}$, which is a weak homotopy equivalence of simplicial sets by virtue of Proposition 3.4.5.4. $\square$