  If there is an elementary embedding from A to B then A and B are elementarily equivalent. On the other hand an embedding between elementarily equivalent structures, or even between isomorphic structures, need not be elementary. (For example, writing Z for the abelian group of the integers with signature consisting of 0 and +, the embedding from Z to Z that takes each integer n to 2n is an embedding, and of course Z is isomorphic to itself; but this embedding is not elementary, since 1 satisfies the formula ¬∃y(y + y = v1), but 2 doesn’t.)