that is both a Grothendieck fibration as well as an opfibration.
Therefore every morphism in a bifibration has both a push-forward as well as a pullback .
If the fibration is even a bifibration, there is a particularly elegant algebraic way to encode its descent properties; this is monadic descent. The Benabou–Roubaud theorem characterizes descent properties for bifibrations.
A bifibration such that is a bifibration as well is called a trifibration (cf. Pavlović 1990, p.315).
Duško Pavlović, Categorical Interpolation: Descent and the Beck-Chevalley Condition without Direct Images , pp.306-325 in Category theory Como 1990, LNM 1488 Springer Heidelberg 1991.