Exercise 5.3.2.17. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets with the following properties:
For every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is a Kan complex.
For every morphism $u: C \rightarrow C'$ in $\operatorname{\mathcal{C}}$, the induced map $\mathscr {F}(u): \mathscr {F}(C) \rightarrow \mathscr {F}(C')$ is a Kan fibration.
Show that the projection map $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a left fibration of simplicial sets.