Construction 11.5.0.33. Let $X$ be a braced simplicial set. Every nondegenerate simplex $\sigma : \Delta ^{n} \rightarrow X$ determines a functor
which we can identify with an $n$-simplex $f_0(\sigma )$ of the simplicial set $\operatorname{Ex}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}}) )$ (Example 3.3.2.8). The construction $\sigma \mapsto f_0(\sigma )$ determines a morphism of semisimplicial sets $f_0: X^{\mathrm{nd}} \rightarrow \operatorname{Ex}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}} ) )$, which extends uniquely to a map of simplicial sets $f: X \rightarrow \operatorname{Ex}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X}^{\mathrm{nd}} ) )$ (Proposition 3.3.1.5).