Warning 2.3.1.11. Let $\operatorname{\mathcal{C}}$ be a $2$-category. By virtue of Proposition 2.3.1.9, we can identify $n$-simplices of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ with triples
satisfying condition $(c')$ of Proposition 2.3.1.9. This gives a description of $\operatorname{N}_{n}^{\operatorname{D}}(\operatorname{\mathcal{C}})$ which makes no reference to the identity $1$-morphisms of $\operatorname{\mathcal{C}}$ or the left and right unit constraints of $\operatorname{\mathcal{C}}$. The resulting identification is functorial with respect to injective maps of linearly ordered sets $[m] \rightarrow [n]$. In other words, we can construct the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ as a semisimplicial set (see Definition 1.1.1.2) without knowing the left and right unit constraints of $\operatorname{\mathcal{C}}$. However, the left and right unit constraints of $\operatorname{\mathcal{C}}$ are needed to define the degeneracy operators on the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$.