Functorial test modules

In this article we introduce a slight modification of the definition of test modules which is an additive functor τ on the category of coherent Cartier modules. We show that in many situations this modification agrees with the usual definition of test modules. Furthermore, we show that for a smooth morphism $f:X\to Y$ of F-finite schemes one has a natural isomorphism $f^{!}°\tau ?\tau °f^{!}$. If f is quasi-finite and of finite type we construct a natural transformation $\tau °f_{?}\to f_{?}°\tau$.