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The Auslander categories with respect to $C$ are related through Foxby equivalence: $\\xymatrix@C=50pt{\\mathcal {A}_C(R) \\ar@<0.4ex>[r]^{C\\otimes^{\\mathbf{L}}_{R} -} & \\mathcal {B}_C(R)\n  \\ar@<0.4ex>[l]^{\\mathbf{R}\\mathrm{Hom}_{R}(C, -)}}$. We firstly intend to extend the Foxby equivalence to Cartan-Eilenberg complexes. To this end, C-E Auslander categories, C-E $\\mathcal{W}$ complexes and C-E $\\mathcal{W}$-Gorenstein complexes are introduced, where $\\mathcal{W}$ denotes a self-orthogonal class of $R$-modules. 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The Auslander categories with respect to $C$ are related through Foxby equivalence: $\\xymatrix@C=50pt{\\mathcal {A}_C(R) \\ar@<0.4ex>[r]^{C\\otimes^{\\mathbf{L}}_{R} -} & \\mathcal {B}_C(R)\n  \\ar@<0.4ex>[l]^{\\mathbf{R}\\mathrm{Hom}_{R}(C, -)}}$. We firstly intend to extend the Foxby equivalence to Cartan-Eilenberg complexes. To this end, C-E Auslander categories, C-E $\\mathcal{W}$ complexes and C-E $\\mathcal{W}$-Gorenstein complexes are introduced, where $\\mathcal{W}$ denotes a self-orthogonal class of $R$-modules. 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