E \to B
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 decent properties; this is monadic descent. The Benabou–Roubaud theorem characterizes descent properties for bifibrations.