Amalgamations of classes of Banach spaces with a monotone basis

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 $C$.