Canonization for disjoint unions of theories

If there exist efficient procedures (canonizers) for reducing terms of two first-order theories to canonical form, can one use them to construct such a procedure for terms of the disjoint union of the two theories? We prove this is possible whenever the original theories are convex. As an application, we prove that algorithms for solving equations in the two theories (solvers) can not be combined in a similar fashion. These results are relevant to the widely used Shostak's method for combining decision procedures for theories. They provide the first rigorous answers to the questions about the possibility of directly combining canonizers and solvers.