  We give a new construction of sheaves on a relative site associated to a
product $X\times S$ where $S$ plays the role of a parameter space, expanding
the previous construction by the same authors, where the subanalytic structure
on $S$ was required. Here we let this last condition fall. In this way the
construction becomes much easier to apply when dimension of $S$ is bigger than
one. We also study the functorial properties of base change with respect to the
parameter space.
