Notation 5.3.3.23. Let $\operatorname{\mathcal{C}}$ be a category and suppose we are given a morphism of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. For every object $C \in \operatorname{\mathcal{C}}$, let $\mathscr {G}_{\operatorname{\mathcal{E}}}(C)$ denote the fiber product $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}$. The construction $C \mapsto \mathscr {G}_{\operatorname{\mathcal{E}}}(C)$ then determines a functor $\mathscr {G}_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$.
Suppose we are given an $n$-simplex $\sigma $ of $\operatorname{\mathcal{E}}$. Then $U( \sigma )$ is an $n$-simplex of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, which we can identify with a diagram
in the category $\operatorname{\mathcal{C}}$. For $0 \leq m \leq n$, we can view the diagram
as an $m$-simplex $\overline{\tau }_ m$ of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{ / C_ m } )$. The pair $( \overline{\tau }_ m, U(\sigma )|_{ \Delta ^ m} )$ can then be viewed as an $m$-simplex $\tau _ m$ of $\mathscr {G}_{\operatorname{\mathcal{E}}}(C_ m)$. Setting $\tau = ( \tau _0, \tau _1, \cdots , \tau _ n )$, we observe that the pair $(U(\sigma ), \tau )$ can be regarded as an $n$-simplex $u_{\operatorname{\mathcal{E}}}( \sigma )$ of the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {G}_{\operatorname{\mathcal{E}}}}(\operatorname{\mathcal{C}})$. Allowing $n$ to vary, the construction $\sigma \mapsto u_{\operatorname{\mathcal{E}}}(\sigma )$ determines a morphism of simplicial sets $u_{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}_{\operatorname{\mathcal{E}}}}(\operatorname{\mathcal{C}})$ for which the diagram
is commutative.