Let \(_RC_S\) be a semidualizing bimodule, where \(R\) and \(S\) are general associative rings, as introduced by H. Holm and D. White [J. Math. Kyoto Univ. 47, No. 4, 781–808 (2007; Zbl 1154.16007)]. This notion generalizes the commutative one introduced by H.-B. Foxby [Math. Scand. 31, 267–284 (1973; Zbl 0272.13009)], E. S. Golod [Tr. Mat. Inst. Steklova 165, 62–66 (1984; Zbl 0577.13008)] and W. V. Vasconcelos [Divisor theory in module categories. Elsevier, Amsterdam (1974; Zbl 0296.13005)]. The authors define \(C\)-weak injective, \(C\)-weak flat modules that extend the class of weak injective and weak flat modules. Recall that a left \(R\)-module \(M\) is super finitely presented if it has a projective resolutions with all terms finitely generated, this modules are called \(FP_\infty\) by other authors as well. A left \(R\)-module \(E\) (resp. right \(R\)-module F) is weak injective (resp. weak flat) if \(\mathrm{Ext}_R^1(X, E)=0\) (resp. \(\mathrm{Tor}_1^R(F, X)=0\) for any super finitely presented left \(R\)-module \(X\). These classes of modules were introduced by Z. Gao and F. Wang [Commun. Algebra 43, No. 9, 3857–3868 (2015; Zbl 1334.16008)]. Then \(C\)-weak flat right \(S\)-module (resp. \(C\)-weak injective right \(R\)-module) has the form \(C\otimes F\) (resp. \(\mathrm{Hom}_S(C,I)\)) for some weak flat module \(F_R\) (resp. weak injective module \(I_S\)). It is shown, in Section 2, that weak flat modules are contained in the Auslander class \({\mathcal A}_C(R)\) and weak injective modules are contained in the Bass class, \({\mathcal G}_C(R)\). The rest of the section is devoted to obtain several properties of \(C\)-weak flat and \(C\)-weak injective modules analogous to the weak notions. Section 3 contains the extension of Foxby equivalence to this setting. In the last section, it is shown that if we iterated the procedure to obtain the Auslander and Bass classes with respect to \(C\), new objects are not obtained.