In SGA I.6 Grothendieck and Pierre Gabriel presented the formalism of fibered categories and descent; the exposition and the terminology emphasises on a version of the relative point of view for categories. Thus they start with preliminaries over the slice 2-category of functors with fixed target category . Objects in are thus called categories over and the morphisms are called functors over . Standard passages between different slice categories are via the base change and the cobase change 2-functors.
An important sub-2-category of is the 2-category of fibered categories over ; this is not a full sub-2-category: one restricts only to cartesian functors among fibered categories. The 2-categories are fibers in a codomain 2-fibration over .
Mike Shulman: Is that really a 2-fibration? You can pull back along along a functor , but can you do anything with a natural transformation ?
Zoran: sorry there is indeed an error there, but I will not yet erase it, as there is (in my memory) a partial repair of the statement for which I need to sit down to formulate precisely.
Its restriction to the sub-2-category of fibered categories is then also a Grothendieck 2-fibration which has been studied by Claudio Hermida.
C. Hermida, Some properties of as a fibred 2-category, Journal of Pure and Applied Algebra 134 (1), 83–109, 1999 (earlier draft version can be found at jpaa.ps.gz)