Corollary 2.3.4.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. Then passage to the Duskin nerve induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Strictly unitary functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} \ar [d] \\ \{ \textnormal{Maps $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$ preserving thin $2$-simplices} \} . } \]