It was recently proved ([12]) that the class of Gorensteininjective left R-modules is both covering and enveloping over a two sided noetherian ring R with the property that the character modules of the Gorenstein injective left R-modules are Gorenstein flat. It was also proved that over the same type of rings, the class of Gorenstein at right R-modules is preenveloping ([11]). We prove here that if R is a two sided noetherian ring R such that R satises the Auslander condition and has nite nitistic left injective dimension then R has the desired property: the character module of any Gorenstein injective is Gorenstein flat.