  Let $S$ and $R$ be rings and $_SC_R$ a (faithfully) semidualizing bimodule.
We introduce and study $C$-weak flat and $C$-weak injective modules as a
generalization of $C$-flat and $C$-injective modules (J. Math. Kyoto Univ.
47(2007), 781-808) respectively, and use them to provide additional information
concerning the important Foxby equivalence between the subclasses of the
Auslander class $\mathcal{A}_C(R)$ and that of the Bass class
$\mathcal{B}_C(S)$. Then we study the stability of Auslander and Bass classes,
which enables us to give some alternative characterizations of the modules in
$\mathcal{A}_C(R)$ and $\mathcal{B}_C(S)$. Finally we consider an open question
which is closely relative to the main results (Proc. Edinb. Math. Soc.
48(2005), 75-90), and discuss the relationship between the Bass class
$\mathcal{B}_C(S)$ and the class of Gorenstein injective modules.
