Variant 7.5.2.4 (Homotopy Limits of Ordinary Categories). Let $\operatorname{Cat}$ denote the (ordinary) category of categories, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Cat}$ be a functor, and let $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the category of elements of $\mathscr {F}$. We let $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ denote the category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$ whose objects are sections of the projection functor $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ which carry each morphism of $\operatorname{\mathcal{C}}$ to a $U$-cocartesian morphism of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Let $\operatorname{N}_{\bullet }( \mathscr {F} )$ denote the $\operatorname{QCat}$-valued functor given by $C \mapsto \operatorname{N}_{\bullet }( \mathscr {F}(C) )$. Combining Proposition 1.5.3.3 with Example 5.6.1.8, we obtain a canonical isomorphism of simplicial sets
In particular, the formation of homotopy limits preserves the full subcategory of $\operatorname{QCat}$ spanned by the (nerves of) ordinary categories.