Remark 5.6.2.5. Let $\operatorname{\mathcal{C}}$ be an ordinary category equipped with a strictly unitary functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$. Then the construction $C \mapsto \operatorname{N}_{\bullet }(\mathscr {F}(C) )$ determines a functor of $\infty $-categories $\operatorname{N}_{\bullet }(\mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$ (see Remark 5.5.4.10). In ยง5.6.3, we will construct a canonical isomorphism
where the simplicial set on the left hand side is given by Definition 5.6.2.1 and $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is the category of elements introduced in Definition 5.6.1.1 (see Proposition 5.6.3.4). Stated more informally, we can regard the $\infty $-category of elements construction (Definition 5.6.2.4) as a generalization of the classical category of elements construction (Definition 5.6.1.1).