Rationalized evaluation subgroups of a map II: Quillen models and adjoint maps

We identify the long exact sequence induced on rational homotopy groups by the evaluation map ω:map(X,Y;f)→Y, and in particular the rationalization of the evaluation subgroups of ff, in terms of derivations of Quillen models and adjoint maps. We consider a generalization of a question of Gottlieb within the context of rational homotopy theory. We also study the rationalization of the GG-sequence of a map. In a separate result of independent interest, we give an explicit Quillen minimal model of a product A×XA×X, in the case in which AA is a rational co-HH-space.