Prove or disprove: Let $\Sigma = (U; f_1,..,f_n)$ be a signature, let $S,R$ be $\Sigma-$structures, let $S$ be unique. Let $\phi: U_S \rightarrow U_R$ be an isomorphism. Then $R$ is unique.
I really don't know how to prove this well and not sure if I got it correctly?
So if we assume that $S$ is unique and $R$ is unique, then they are abstractly the same in that one is a copy of the other in the sense that there is a one one correspondence with gives homomorphisms both ways between $R$ and $S$ which means that both structures are isomorphic. Task assumed that we have the isomorphism $\phi: U_S \rightarrow U_R$ and thus $R$ is really unique and the statement is true...?