Corollary 5.3.6.8. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a cocartesian fibration of simplicial sets and let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Then:
- $(1)$
The projection map $\pi : \operatorname{Fun}( \operatorname{\mathcal{C}}/ \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$ is a cartesian fibration of simplicial sets.
- $(2)$
Let $e$ be an edge of the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}/ \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$, corresponding to a pair $( \overline{e}, f )$, where $\overline{e}$ is an edge of the simplicial set $\operatorname{\mathcal{B}}$ and $f: \Delta ^1 \times _{ \operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories. Let $U_{\overline{e}}: \Delta ^1 \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^1$ be given by projection onto the first factor. Then $e$ is $\pi $-cartesian if and only if the functor $f$ carries $U_{\overline{e}}$-cocartesian morphisms to isomorphisms in the $\infty $-category $\operatorname{\mathcal{D}}$.