Kerodon

Notation 11.10.1.1. Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be cartesian fibrations of simplicial sets. We let $\operatorname{Fun}_{/S}^{\operatorname{Cart}}(X,X')$ denote the full subcategory of $\operatorname{Fun}_{/S}(X,X')$ spanned by those morphisms $f: X \rightarrow X'$ which carry $q$-cartesian edges of $X$ to $q'$-cartesian edges of $X'$. If $q: X \rightarrow S$ and $q': X' \rightarrow S$ are cocartesian fibrations, we let $\operatorname{Fun}_{/S}^{\operatorname{CCart}}(X,X')$ denote the full subcategory of $\operatorname{Fun}_{/S}(X,X')$ spanned by those morphisms $f: X \rightarrow X'$ which carry $q$-cocartesian edges of $X$ to $q'$-cocartesian edges of $X'$.