Foliated categories (catégories feuilletées) were introduced by Jean Benabou in an ongoing work which started in 1984 and is yet not published. They are a weaker structure than a fibered category, but still allow one to test for various standard properties of functors fibrewise.

The definition goes as follows: a functor $P\colon \mathbf{X} \to \mathbf{B}$ makes $\mathbf{X}$ a foliated category (over $\mathbf{B}$) if every morphism $f$ in $\mathbf{X}$ factors as a $P$-vertical morphism $v$ (i.e. $P(v)$ is an identity morphism in $\mathbf{B}$), followed by a $P$-cartesian morphism$k$, i.e. $f = k\circ v$, and, further, $P$-cartesian morphisms are closed under composition. (Note that this is the weaker notion of cartesian morphism.)

A morphism between foliated categories $\mathbf{X}'\to \mathbf{X}$ (over $\mathbf{B}$) is a functor over $\mathbf{B}$ that sends cartesian morphisms to cartesian morphisms, and such that for every object $X'$ of $\mathbf{X}'$, and morphism $f\colon Y\to F(X')$ in $\mathbf{X}$, there is a factorisation $f = F(k)\circ v$, where $v$ is vertical in $\mathbf{X}$ and $k$ is cartesian in $\mathbf{X}'$. Bénabou calls such functors cartesian.

References

Jean Bénabou, Foncteurs cartésiens et catégories feuilletées (9 novembre 2012, journée Guitart, Paris) YouTubeslides

Jean Bénabou, Du vieux et du neuf sur la construction de Grothendieck, March 2019, YouTube (the material on foliated categories—called simply foliations—starts after timestamp 1:01:00).