Given (possibly weak) 2-categories, and , and (possibly weak) 2-functors and , a biadjunction is given by specifying for each object in and each object in an equivalence of categories , which is pseudonatural both in and in .
There are several other characterizations of biadjointness.
If there is a biadjunction in this sense, it can be replaced by a biadjunction for which this equivalence of categories is an adjoint equivalence.
John Gray?, Formal category theory: Adjointness for 2-categories, Lecture Notes in Mathematics 391, Springer, Berlin, 1974.
Thomas M. Fiore, Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory, Memoirs of the American Mathematical Society 182 (2006), no. 860. 171 pages, MR2007f:18006, math.CT/0408298