Example 5.3.3.9 (The Weighted Nerve of a Constant Diagram). Let $\operatorname{\mathcal{C}}$ be a category, let $X$ be a simplicial set, and let $\underline{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the constant functor taking the value $X$. Then Remark 5.3.3.7 and Example 5.3.3.2 supply an isomorphism of simplicial sets $\operatorname{N}_{\bullet }^{\underline{X}}(\operatorname{\mathcal{C}}) \simeq X \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
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