# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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 $\operatorname{\mathcal{D}}$.

Proof. Apply Proposition 5.3.6.6 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$ (and use Example 5.1.1.4). $\square$