Stability of Gorenstein flat categories with respect to a semidualizing module

In this paper, we first introduce ${\mathcal{W}}_F$-Gorenstein modules to establish the following Foxby equivalence: $\xymatrix@C=80pt{\mathcal {G}(\mathcal {F})\cap \mathcal {A}_C® \ar@<0.5ex>[r]^{C\otimes_R-} & \mathcal {G}(\mathcal {W}_F) \ar@<0.5ex>[l]^{$\text{Hom}$_R(C,-)}} $where$\mathcal {G}(\mathcal {F})$,$\mathcal {A}_C® $and$\mathcal {G}(\mathcal {W}F)$ denote the class of Gorenstein flat modules, the Auslander class and the class of ${\mathcal{W}}_F$-Gorenstein modules respectively. Then, we investigate two-degree ${\mathcal{W}}_F$-Gorenstein modules. An $R$-module $M$ is said to be two-degree ${\mathcal{W}}_F$-Gorenstein if there exists an exact sequence $$\mathbb{G}$\bullet=\indent …\longrightarrow G_1\longrightarrow G_0\longrightarrow G^0\longrightarrow G^1\longrightarrow…$in$\mathcal {G}(\mathcal {W}_F)$suchthat$M \cong$ $\text{im}(G_0\to G^0)$ and that ${\mathbb{G}}_{\bullet }$ is Hom${}_R(\mathcal{G}({\mathcal{W}}_F),-)$ and $\mathcal {G}(\mathcal {W}_F)^+\otimes_R-$ exact. We show that two notions of the two-degree ${\mathcal{W}}_F$-Gorenstein and the ${\mathcal{W}}_F$-Gorenstein modules coincide when R is a commutative GF-closed ring.