# Kerodon

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Proposition 5.3.1.7. 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, for every object $C \in \operatorname{\mathcal{C}}$, the evaluation map of Remark 5.3.1.3 induces a trivial Kan fibration of $\infty$-categories $\operatorname{ev}_{C}: \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) \rightarrow \operatorname{\mathcal{E}}_{C}$.

Proof of Proposition 5.3.1.7. 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. By virtue of Example 5.3.1.14, it will suffice to show that the evaluation functor

$\operatorname{ev}_{C}: \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}^{\operatorname{CCart}}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}_{C}$

is a trivial Kan fibration. This is a special case of Corollary 5.3.1.22, since the identity morphism $\operatorname{id}_{C}$ is initial when viewed as an object of the coslice category $\operatorname{\mathcal{C}}_{C/}$. $\square$