Kerodon

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Corollary 5.3.3.16. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is a Kan complex. Then the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a left fibration.

Proof. Proposition 5.3.3.15 guarantees that $U$ is a cocartesian fibration. Moreover, for every object $C \in \operatorname{\mathcal{C}}$, the fiber $U^{-1} \{ C\} \simeq \mathscr {F}(C)$ is a Kan complex (Example 5.3.3.8). Applying Proposition 5.1.4.14, we conclude that $U$ is a left fibration. $\square$