The dual fibration of a (Grothendieck) fibration is a fibration over the same base but with all of the fibers dualized.
Let be a fibration, and let be the corresponding indexed category. The dual fibration can be defined (Borceux) as the fibration associated (via the Grothendieck construction) to the composite
where is the operation sending any category to its opposite (note this operation preserves the direction of the 1-cells, although it reverses the direction of the 2-cells in Cat). (See Borceux reference.) Alternatively, the dual fibration may be defined in more elementary terms as a category with the same objects as , and whose morphisms are equivalence classes of spans where is vertical (i.e., ) and is horizontal (i.e., -cartesian). (See Pavlovic and Kock references.)
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. (pdf)
Francis Borceux, Handbook of Categorical Algebra (see Volume 2, 8.3).
Anders Kock, The dual fibration in elementary terms, arXiv:1501.01947.
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