The Schröder-Bernstein property for weakly minimal theories

For a countable, weakly minimal theory T , we show that the SchröderBernstein property (any two elementarily bi-embeddable models are isomorphic) is equivalent to each of the following: 1. For any U -rank-1 type q ∈ S(acl(∅)) and any automorphism f of the monster model C, there is some n < ω such that f(q) is not almost orthogonal to q ⊗ f(q)⊗ . . .⊗ fn−1(q); 2. T has no infinite collection of models which are pairwise elementarily bi-embeddable but pairwise nonisomorphic. We conclude that for countable, weakly minimal theories, the SchröderBernstein property is absolute between transitve models of ZFC.