Corollary 2.3.5.7. For every strict $2$-category $\operatorname{\mathcal{C}}$, there is a canonical isomorphism of simplicial sets
\[ \operatorname{Sing}_{\bullet }^{\operatorname{Path}_{(2)}[\bullet ] }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}), \]
given on $n$-simplices by composition with the lax functor $T_{[n]}: [n] \rightarrow \operatorname{Path}[n]$ of Construction 2.3.5.4. In other words, the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is given by
\[ \operatorname{N}^{\operatorname{D}}_{n}(\operatorname{\mathcal{C}}) \simeq \{ \textnormal{Strict functors $\operatorname{Path}_{(2)}[n] \rightarrow \operatorname{\mathcal{C}}$} \} . \]