Kerodon

Remark 2.3.1.12. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a lax functor. If $F$ is strictly unitary, then composition with $F$ induces a map of simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{D}})$. However, even without the assumption that $F$ is strictly unitary, one can use the description of Proposition 2.3.1.9 to obtain a collection of maps $\operatorname{N}^{\operatorname{D}}_{n}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{N}^{\operatorname{D}}_{n}(\operatorname{\mathcal{D}})$ which are compatible with the face operators on the simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ and $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$ (though not necessarily with the degeneracy operators). In other words, if we regard the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ as a semisimplicial set, then it is functorial with respect to all (lax) functors between $2$-categories.