# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary 3.3.1.11. Let $X_{\bullet }$ be a braced simplicial set. Then the inclusion of semisimplicial sets $g_0: X_{\bullet }^{\mathrm{nd}} \hookrightarrow X_{\bullet }$ extends uniquely to an isomorphism $g: ( X_{\bullet }^{\mathrm{nd} })^{+} \simeq X_{\bullet }$.

Proof. It follows from Corollary 3.3.1.9 that $g_0$ extends uniquely to a map of simplicial sets $g: ( X_{\bullet }^{\mathrm{nd}} )^{+} \rightarrow X_{\bullet }$. To show that $g$ is an isomorphism, it will suffice to show that for every simplicial set $Y_{\bullet }$, composition with $g$ induces a bijection

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( X_{\bullet }, Y_{\bullet } ) & \rightarrow & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( (X_{\bullet }^{\mathrm{nd}})^{+}, Y_{\bullet } ) \\ & \simeq & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}, \operatorname{Set})}( X_{\bullet }^{\mathrm{nd}}, Y_{\bullet } ), \end{eqnarray*}

which is precisely the content of Proposition 3.3.1.5. $\square$