# Kerodon

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

Corollary 5.3.4.18. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then there exists a cocartesian fibration of $\infty$-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and a scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$.

Proof. Using Proposition 4.1.3.2, we can choose a diagram of $\infty$-categories $\mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ and a levelwise categorical equivalence $\alpha : \mathscr {F} \rightarrow \mathscr {F}'$. We can then take $\lambda$ to be the composition $\underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \xrightarrow { \alpha } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}' ) \xrightarrow {\lambda _{t}} \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}})$, where $\lambda _{t}$ is the taut scaffold of Proposition 5.3.4.17. $\square$