Complete intersection dimensions and Foxby classes

Using Blanco and Majadas' version of complete intersection dimension for local ring homomorphisms, we prove the following generalization of a theorem of Avramov and Foxby: Given local ring homomorphisms φ:R→S and ψ:S→T such that φ has finite Gorenstein dimension, if ψ has finite complete intersection dimension, then the composition ψ∘φ has finite Gorenstein dimension. This follows from our result stating that, if M has finite complete intersection dimension, then M is C-reflexive and is in the Auslander class AC(R) for each semidualizing R-complex C.