In this paper a localization (germination) process with change of the base space is presented. The data consist of two topological spaces T and S, a continuous function ' : T ! S, a surjective function p : E ! T , a directed family (di)i2I of bounded pseudometrics for p generating a Hausdor uniformity and a family of global selections for p. In terms of these data, a uniform bundle is constructed over the base space S, whose b ers are colimits in a category of uniform spaces. Similar results follow for the case of sheaves of sets. This localization process leads to a universal arrow in a context described in terms of a category of uniform bundles.