Kerodon

Remark 2.3.1.6 (Functoriality). The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ determines a functor from the category $\operatorname{2Cat}_{\operatorname{ULax}}$ of small $2$-categories (with morphisms given by strictly unitary lax functors) to the category $\operatorname{Set_{\Delta }}$ of simplicial sets. This functor fits into the general paradigm of Variant 1.2.2.8: it arises from a cosimplicial object of the category $\operatorname{2Cat}_{\operatorname{ULax}}$, given by the inclusion $\operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Cat}\hookrightarrow \operatorname{2Cat}_{\operatorname{ULax}}$. Beware that, unlike the usual nerve functor $\operatorname{N}_{\bullet }: \operatorname{Cat}\rightarrow \operatorname{Set_{\Delta }}$, the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }: \operatorname{2Cat}_{\operatorname{ULax}} \rightarrow \operatorname{Set_{\Delta }}$ does not admit a left adjoint: Proposition 1.2.3.15 does not apply, because the category $\operatorname{2Cat}_{\operatorname{ULax}}$ does not admit small colimits (one can address this problem by restricting to strict $2$-categories: we will return to this point in §2.3.5).