2-categorical adjunctions for Fourier-Mukai kernels

I will discuss a new approach which should allow one to naturally lift to the level of Fourier-Mukai kernels many useful structures which exist on the level of exact functors between derived categories of schemes: 2-categorical adjunctions, pseudofunctoriality, base change, and projection formula isomorphisms, etc. We obtain explicit and easy to understand formulas which can be readily used in computations.