Kerodon

Proof. Let $n$ be a nonnegative integer. For every semisimplicial set $S$, Example 3.3.3.13 allows us to identify $n$-simplices of the nerve $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}})$ with the set of pairs $(\tau , \sigma )$, where $\tau $ is a $m$-simplex of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}} )$ (given by a diagram of increasing functions $[k_0] \hookrightarrow [k_1] \hookrightarrow \cdots \hookrightarrow [k_ n]$) and $\sigma $ is an element of the set $S_{k_ n}$. It follows that the functor $S \mapsto \operatorname{N}_{n}( \operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}})$ preserves colimits. Allowing $n$ to vary, we conclude that the functor $S \mapsto \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{S^{+}}^{\mathrm{nd}})$ preserves colimits. $\square$