Are isomorphisms between finitely generated substructures determined on a set of generators?

Question

Let $K$ be a structure, $\varphi: A \to B$ be an isomorphism of finitely generated substructures of $K$. Let $a_0,\dots,a_n$ be generators of $A$. Do the images $\varphi(a_i)$ of the generators $a_i$ uniquely determine the isomorphism?

Answer 1

Yes. More generally the following is true:

If a structure $\mathfrak A$ is generated by $S$, then every homomorphism $h:\mathfrak A\to\mathfrak B$ is determined by its values on $S$.

This is Lemma 1.1.7 in Tent & Ziegler "A course in Model Theory". The proof is straight forward and also present in the book.