Approximation factor preserving reduction
That looks right.  The only exception I can think of is problems where there isn't an optimal solution, but just a sequence of ever-better solutions converging to, but never attaining, some optimum.  Though actually, even then, if the optima weren't the same, it should be possible to find some solution in between the optima, which would arrive at a contradiction.  So it seems what you say should still hold, unless I'm also missing something.