Definition [efr-002U]

Let p: \mathcal {D} \to \mathcal {C} be a homomorphism of bicategories. We say p is

1. A (-,-)-fibration if it is a weak 2-fibration in the sense of Buckley
2. A (+,-)-fibration if the 1-cell dual \mathcal {C}^\mathrm {op} \to \mathcal {D}^\mathrm {op} is a (-,-)-fibration. Recall that \mathcal {C}^\mathrm {op}(X,Y) = \mathcal {C}^\mathrm {op}(Y,X), i.e \mathcal {C}^\mathrm {op} inverts the direction of the 1-cells but not the 2-cells.
3. A (-,+)-fibration if the 2-cell dual \mathcal {C}^\mathrm {co} \to \mathcal {D}^\mathrm {co} is a (-,-)-fibration. Recall that \mathcal {C}^\mathrm {co}(X,Y) = \mathcal {C}(X,Y)^\mathrm {op}
4. A (+,+)-fibration if the bidual \mathcal {C}^{\mathrm {co} \mathrm {op}} \to \mathcal {D}^{\mathrm {co} \mathrm {op}} is a (-,-)-fibration.

Note that the way these correspond to functors from the various duals of \mathcal {D} into bicategories is somewhat unintuitive! By the work of Buckley, (-,-)-fibrations classify functors \mathcal {D}^{\mathrm {co} \mathrm {op}} \to \mathsf {BiCat}. Given for example a (+,-)-fibration, it has a dual (-,-)-fibration which, of course, classifies a functor \mathcal {D}^\mathrm {co} \to \mathsf {BiCat}. But the values of this functor are not the fibers of the original functor, but the (1-cell) duals---after postcomposing with the (-)^\mathrm {op} functor \mathsf {BiCat}^\mathrm {co} \to \mathsf {BiCat}, we are left with a functor \mathcal {D} \to \mathsf {BiCat}.