Excercise 5.9, p.44 Ebbinghaus et al - Mathematical logic:
This excercise was discussed here: Can a single sentence be used to distinguish between isomorphic classes of finite structures?
Roughly speaking, we can find a sentence phi which incorporates all rules given by structure A (existence and uniqueness of all elements of the underlying domain, by A defined functions, by A defined constants, by A defined relations), so that A and an isomorphic structures B satisfies phi.
It is asserted that the converse is true, either, i.e. the structures satisfying phi are precisely those isomorphic to A. But what if I take A and extend A by another function/constant/relation and thereby obtain a structure B.
The sentence phi is still satisfied by B but A and B are not isomorphic. I guess I am missing/forgetting some detail of a definition or so.