Let $R$ be a commutative noetherian ring with a semi-dualizing module $C$. The Auslander categories with respect to $C$ are related through Foxby equivalence: $xymatrix@C=50pt{mathcal {A}_C(R) ar@<0.4ex>[r]^{Cotimes^{mathbf{L}}_{R} -} & mathcal {B}_C(R) ar@<0.4ex>[l]^{mathbf{R}mathrm{Hom}_{R}(C, -)}}$. 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. Moreover, criteria for finiteness of C-E Gorenstein dimensions of complexes in terms of resolution-free characterizations are considered.
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