Summary: In order to establish the Foxby equivalences associated to strongly Gorenstein modules, we introduce the notions of strongly \(\mathcal{W}_P\)-Gorenstein, \(\mathcal{W}_I\)-Gorenstein and \(\mathcal{W}_F\)-Gorenstein modules and discuss some basic properties of these modules. We show that the subcategory of strongly Gorenstein projective left \(R\)-modules in the left Auslander class and the subcategory of strongly \(\mathcal{W}_P\)-Gorenstein left \(S\)-modules are equivalent under Foxby equivalence. The injective and flat case are also studied.