Example 8.1.3.13. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts and let $n$ be a nonnegative integer. Applying Construction 8.1.3.12 in the special case where $\operatorname{\mathcal{D}}= [n]$, we obtain a function
Using Propositions 1.3.3.1 and 8.1.1.10, we can identify the left hand side with the collection of $n$-simplices of the simplicial set $\operatorname{Cospan}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$. This construction depends functorially on $n$, and therefore determines a morphism of simplicial sets from $\operatorname{Cospan}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$ to the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Cospan}(\operatorname{\mathcal{C}}) )$.