Jacobs defines Cartesian morphisms and fibrations in Definition 1.1.3 of this document.
In other places in the text he uses phrases like "the universal property of this lifting." This made me wonder, can Cartesian morphisms or fibrations be defined in terms of universal morphisms ? If so, I'd be very interested to know how.
Given a functor $p\colon E\to B$, for each object $Y$ it has an associated functor $p/Y\colon E/Y\to B/pY$ from the category whose objects are morphisms in $E$ with codomain $Y$ and whose morphisms are commutative triangles in $E$, to the category whose objects are morphisms in $B$ with codomain $pY$ and whose morphisms are commutative triangles in $B$.
Given a morphism $g\colon Z\to Y$, considered as an object of $E/Y$, a morphism $w\colon (p/Y)g\to u$ in $B/pY$ is a morphism $w\colon pZ\to I$ such that $u\circ w=pg$.
A morphism $k\colon (p/Y)f\to u$ is then universal if for every other morphism $w\colon (p/Y)g\to u$ there is a unique morphism $h\colon Z\to X$ such that $w=k\circ (p/Y)h$.
We see then that $f\colon X\to Y$ satisfying $pf=u$ is Cartesian precisely when the identity morphism $\mathrm{id}_u\colon (p/Y)f\to u$ from the functor $p/Y\colon E/Y\to B/pY$ to $pf$ is universal.
P.S. The notion of a Grothendieck fibration is not stable under equivalence of categories. Functors equivalent to Grothendieck fibrations are the so-called Street fibrations, and can be characterizes by the condition that every object $u$ in $B/pY$ admits a universal morphism $(p/Y)f\to u$ that is an isomorphism.