Example 5.3.2.2 (Discrete Diagrams). Let $\operatorname{\mathcal{C}}$ be a category having only identity morphisms, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ can be identified with the disjoint union ${\coprod }_{C \in \operatorname{\mathcal{C}}} \mathscr {F}(C)$.
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