Example 2.3.1.14 (Edges of the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Using Proposition 2.3.1.9, we can identify edges of the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ with $1$-morphisms $f: X \rightarrow Y$ of the $2$-category $\operatorname{\mathcal{C}}$. Under this identification, the face and degeneracy operators
\[ d^{1}_{0}, d^{1}_1: \operatorname{N}^{\operatorname{D}}_{1}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\operatorname{D}}_{0}(\operatorname{\mathcal{C}}) \quad \quad s^{0}_0: \operatorname{N}^{\operatorname{D}}_{0}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\operatorname{D}}_{1}(\operatorname{\mathcal{C}}) \]
are given by $d^{1}_0(f: X \rightarrow Y) = Y$, $d^{1}_1( f: X \rightarrow Y) = X$, and $s^{0}_0(X) = \operatorname{id}_ X$.