Example 5.3.2.5 (Set-Valued Functors). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a diagram of sets indexed by $\operatorname{\mathcal{C}}$. Let us abuse notation by identifying $\mathscr {F}$ with a diagram of simplicial sets (by identifying each of the sets $\mathscr {F}(C)$ as a discrete simplicial set). Then there is a canonical isomorphism of simplicial sets
\[ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \simeq \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F} ). \]
Here $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denotes the category of elements of of the functor $\mathscr {F}$ (Construction 5.2.6.1).