Variant 5.3.3.26. Let $\operatorname{\mathcal{C}}$ be a category, and let us regard $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ as equipped with the simplicial enrichment described in Example 2.4.2.2. For every morphism of simplicial sets $\operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and every functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, precomposition with the morphism $u_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}_{\operatorname{\mathcal{E}}} }(\operatorname{\mathcal{C}})$ of Notation 5.3.3.23 induces an isomorphism of simplicial sets
To see that this map is bijective on $m$-simplices, we can replace $\operatorname{\mathcal{E}}$ by the product $\Delta ^ m \times \operatorname{\mathcal{E}}$ to reduce to the case $m = 0$, in which case it follows from Proposition 5.3.3.24.