Cartesian morphisms and fibrations

Definition 23.1.8. For a functor \(p\colon E \to C\), we say that a morphism \(\phi \colon e \to e'\) in \(E\) is \(p\)-cartesian if for every third object \(e''\) in \(E\) the commutative square

is a pullback square. We say \(p\) is a cartesian fibration if for every morphism \(f\colon x \to y\) and every \(e' \in E_y\) there exists a \(p\)-cartesian lift \(\phi \colon e \to e'\) of \(f\). We similarly obtain a notion of a cartesian functor over \(C\), and this leads to a (non-full) subcategory \[ \Cart (C) \subseteq (\Cat _{\infty })_{/C}. \]