Proposition 5.3.4.8. Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty $-categories. Then the morphism $\lambda _{u}: \underset { \longrightarrow }{\mathrm{holim}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \rightarrow \operatorname{\mathcal{E}}$ of Construction 5.3.4.7 is a scaffold, in the sense of Definition 5.3.4.2.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. It is clear that the composition $U \circ \lambda _{u}$ coincides with the projection map $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Let $e$ be a horizontal edge of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} )$, determined by a morphism $\overline{e}: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$ together with a degenerate edge $\operatorname{id}_{T}$ of the simplicial set $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}( C)$. Identifying $T$ with an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C / } ), \operatorname{\mathcal{E}})$, we see that $\lambda _{u}(e)$ coincides with the morphism $T( \overline{e} )$ and is therefore a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$. To complete the proof, we observe that for every object $C \in \operatorname{\mathcal{C}}$, the induced map
\[ \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \xrightarrow {\lambda } \{ C \} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}} \]
agrees with the map $\operatorname{ev}_{C}: \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C / } ), \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}_{C}$ given by evaluation on the initial object $\operatorname{id}_{C} \in \operatorname{\mathcal{C}}_{C/}$, and is therefore a trivial Kan fibration of simplicial sets (Proposition 5.3.1.7). $\square$