Given two structures (in the model theory sense), I believe the existence of an isomorphism between them implies that they are elementarily equivalent. The implication of elementary equivalence is what I understand to be the most important property of isomorphisms, so when I'm attempting to understand natural isomorphisms from category theory, I'm looking for a similar implication.
Does a natural isomorphism between functors imply that the functors satisfy the same set of first order sentences?
Does my understanding of the importance of isomorphisms between structures make sense?
If natural isomorphisms have nothing to do with anything like elementary equivalence, then what makes them important?