  It was proved by Argyros and Dodos that, for many classes $ C $ of separable
Banach spaces which share some property $ P $, there exists an isomorphically
universal space that satisfies $ P $ as well. We introduce a variant of their
amalgamation technique which provides an isometrically universal space in the
case that $ C $ consists of spaces with a monotone Schauder basis. For example,
we prove that if $ C $ is a set of separable Banach spaces which is analytic
with respect to the Effros-Borel structure and every $ X \in C $ is reflexive
and has a monotone Schauder basis, then there exists a separable reflexive
Banach space that is isometrically universal for $ C $.
