$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark (Edges of the Weighted Nerve). Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, and let $(C,x)$ and $(D,y)$ be vertices of the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ (see Remark Edges of the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ with source $(C,x)$ and target $(D,y)$ can be identified with pairs $(f, e)$, where $f: C \rightarrow D$ is a morphism of the category $\operatorname{\mathcal{C}}$ and $e: \mathscr {F}(f)(x) \rightarrow y$ is an edge of the simplicial set $\mathscr {F}(D)$.