  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: $\xymatrix@C=50pt{\mathcal {A}_C(R)
<EMAIL_ADDRESS>-} & \mathcal {B}_C(R)
  \ar@<0.4ex>[l]^{\mathbf{R}\mathrm{Hom}_{R}(C, -)}}$. 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.
