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Example ($2$-Simplices of the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Using Proposition, we see that a $2$-simplex $\sigma $ of the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with the following data:

  • A triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$.

  • A triple of $1$-morphisms $f: X \rightarrow Y$, $g: Y \rightarrow Z$, and $h: X \rightarrow Z$ in the $2$-category $\operatorname{\mathcal{C}}$ (corresponding to the facts $d_2(\sigma )$, $d_0(\sigma )$, and $d_1(\sigma )$, respectively).

  • A $2$-morphism $\mu : g \circ f \Rightarrow h$, which we depict as a diagram

    \[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{\mu } & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z. } \]