Variant 5.3.1.11 (Cartesian Sections). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be cartesian fibrations of simplicial sets. We let $\operatorname{Fun}^{\operatorname{Cart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$ denote the full subcategory of $\operatorname{Fun}_{/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$ whose objects are morphisms $F: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$ which satisfy the identity $U \circ F = U'$ and carry $U'$-cartesian edges of $\operatorname{\mathcal{E}}'$ to $U$-cartesian edges of $\operatorname{\mathcal{E}}$. Note that we have a canonical isomorphism of simplicial sets
In the special case $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}$, we will refer to $\operatorname{Fun}^{\operatorname{Cart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ as the $\infty $-category of cartesian sections of $U$.