nLab
pseudoadjunction

• 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
• J.W. Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics 391, Springer, Berlin, 1974.
• I. J. Le Creurer, F. Marmolejo, E. M. Vitale, Beck’s theorem for pseudo-monads, J. Pure Appl. Algebra 173 (2002), no. 3, 293–313, MR2003f:18004, doi