Given a real analytic manifold Y, denote by $Y_{sa}$ the associated subanalytic site. Now consider a product Y = X × S. We construct the endofunctor $ℱ\mapsto {ℱ}^S$ on the category of sheaves on $Y_{sa}$ and study its properties. Roughly speaking, ${ℱ}^S$ is a sheaf on $X_{sa}\times S$. As an application, one can now define sheaves of functions on Y which are tempered or Whitney in the relative sense, that is, only with respect to X.