Unique isomorphism between finitely generated $\tau$-structures

Question

Let $\mathcal{A}$, $\mathcal{B}$ be two $\tau$-structures, where $\tau$ is a first order language and let $\vec{a} = (a_1, \dots, a_n) \in A^n$ and $\vec{b} = (b_1, \dots, b_n) \in B^n$. We denote $\langle \vec{a}\rangle$ to be the smallest $\tau$-substructure of $\mathcal{A}$ containing $\{a_1, \dots, a_n\}$ (same definition for $\langle \vec{b} \rangle$). Suppose $\xi: \langle \vec{a} \rangle \to \langle \vec{b} \rangle$ is an isomorphism of $\tau$-structures such that $\xi(a_i) = b_i$ for $i=1, \dots, n$. Is $\xi$ unique in respect to these properties? If so, how can I show it? It seems to me to be unique, because if two isomorphisms agree in a set of generators they should (I don't knou how to prove it) be equal.

Answer 1

Suppose $\xi$ and $\xi'$ are two such isomorphisms. Let $S=\{x:\xi(x)=\xi'(x)\}$. Note that $S$ is a $\tau$-substructure of $\langle \vec{a}\rangle$ since $\xi$ and $\xi'$ are both homomorphisms, and $a_i\in S$ for each $i$ by hypothesis. So by definition of $\langle \vec{a}\rangle$, $S$ must be all of $\langle \vec{a}\rangle$.