  Several results related to flat Friedmann-Lema\^{\i}tre-Robertson-Walker
models in the conformal (Einstein) frame of scalar-tensor gravity theories are
extended. Scalar fields with arbitrary (positive) potentials and arbitrary
coupling functions are considered. Mild assumptions under such functions
(differentiable class, number of singular points, asymptotes, etc) are
introduced in a straightforward manner in order to characterize the asymptotic
structure on a phase space. We pay special attention to the possible scaling
solutions. Numerical evidence confirming our results is presented.
