Example 2.3.1.16 ($3$-Simplices of the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Using Proposition 2.3.1.9, we see that a map of simplicial sets $\partial \Delta ^3 \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with the following data:
A collection of objects $\{ X_ i \} _{0 \leq i \leq 3}$ of the $2$-category $\operatorname{\mathcal{C}}$.
A collection of $1$-morphisms $\{ f_{j,i}: X_ i \rightarrow X_ j \} _{0 \leq i < j \leq 3}$.
A quadruple of $2$-morphisms
\[ \mu _{2,1,0}: f_{2,1} \circ f_{1,0} \Rightarrow f_{2,0} \quad \quad \mu _{3,2,1}: f_{3,2} \circ f_{2,1} \Rightarrow f_{3,1} \]\[ \mu _{3,1,0}: f_{3,1} \circ f_{1,0} \Rightarrow f_{3,0} \quad \quad \mu _{3,2,0}: f_{3,2} \circ f_{2,0} \Rightarrow f_{3,0}. \]
This data can be conveniently visualized as a pair of diagrams
representing “front” and “back” perspectives of the boundary of a $3$-simplex. A $3$-simplex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ can be identified with a map $\operatorname{\partial }\Delta ^3 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ as above which satisfies an additional compatibility condition: namely, the commutativity of the diagram
in the ordinary category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X_0, X_3)$.