Example 3.3.3.12. Let $S$ be a semisimplicial set, and let $S^{+}$ be the braced simplicial set given by Construction 3.3.1.6. Then the category of nondegenerate simplices $\operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}}$ can be described concretely as follows:
The objects of $\operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}}$ are pairs $( [n], \sigma )$, where $[n]$ is an object of $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ and $\sigma $ is an element of $S_{n}$.
A morphism from $([n], \sigma )$ to $([n'], \sigma ')$ in $\operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}}$ is a strictly increasing function $\alpha : [n] \hookrightarrow [n']$ satisfying $\sigma = \alpha ^{\ast }( \sigma ' )$ in the set $S_{n}$.
In other words, $\operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}}$ is the category of elements of the functor $S: \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ (see Variant 5.2.6.2).