  We prove a structural property of the class of unconditionally saturated
separable Banach spaces. We show, in particular, that for every analytic set
$\aaa$, in the Effros-Borel space of subspaces of $C[0,1]$, of unconditionally
saturated separable Banach spaces, there exists an unconditionally saturated
Banach space $Y$, with a Schauder basis, that contains isomorphic copies of
every space $X$ in the class $\aaa$.
