A biadjunction is the “maximally weak” kind of 2-adjunction: a higher generalization of the notion of adjunction from category theory to 2-category theory, and specifically to bicategories (or more generally internally in a tricategory. See 2-adjunction for other kinds of 2-adjunction.
Given (possibly weak) 2-categories, $A$ and $C$, and (possibly weak) 2-functors $F:A\to C$ and $U:C\to A$, a biadjunction is given by specifying for each object $a$ in $A$ and each object $c$ in $C$ an equivalence of categories $C(F a,c)\cong A(a,U c)$, which is pseudonatural both in $a$ and in $c$.
By the Yoneda lemma for bicategories, this is equivalent to giving pseudonatural transformations $\eta : Id_A \to U F$ and $\varepsilon : F U \to Id_C$ satisfying the triangle identities up to invertible modifications. This latter definition can be internalized in any (weak) 3-category, such as a Gray-category or a tricategory.
Both the “equivalence of hom-categories” definition and the “unit and counit” definition have stronger “coherent” versions: We can ask the equivalences $C(F a,c)\cong A(a,U c)$ to be adjoint equivalences, or for the “triangulator” modifications to satisfy the swallowtail equations?. These two conditions are roughly equivalent, and any “incoherent” biadjunction can be improved to a coherent one by altering one of the triangulators; see Gurski, Riehl-Verity, Pstrągowski, Riehl-Shulman, and this proof in Globular.
biadjunction, lax 2-adjunction, pseudoadjunction
biequivalence, biadjoint biequivalence?
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
Nick Gurski, Biequivalences in tricategories, Theory and applications of categories 2012, journal web site, arxiv