  We prove that if $ C $ is a family of separable Banach spaces which is
analytic with respect to the Effros-Borel structure and none member of $ C $ is
isometrically universal for all separable Banach spaces, then there exists a
separable Banach space with a monotone Schauder basis which is isometrically
universal for $ C $ but still not for all separable Banach spaces. We also
establish an analogous result for the class of strictly convex spaces.
