Kerodon

Remark 5.2.1.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, so that the projection map $q: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ is a cartesian fibration (Example 5.1.4.3). In this case, part $(1)$ of Theorem 5.2.1.1 is equivalent to the assertion that for every simplicial set $B$, the simplicial set $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$ is also an $\infty $-category (Theorem 1.5.3.7). By virtue of Proposition 5.1.4.12, part $(2)$ is equivalent to the assertion that a morphism of $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$ is an isomorphism if and only if, for every vertex $b \in B$, its image under the evaluation functor $\operatorname{ev}_{b}: \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is an isomorphism in $\operatorname{\mathcal{C}}$ (Theorem 4.4.4.4).