If $M$ and $N$ are two $L$-structures, and $f: M \rightarrow N$ is an elementary extension, then given any subset $A$ of $M$, $f$ induces in a natural way a morphism $S^M_n(A) \rightarrow S^N_n(f(A))$ of type spaces which is in fact continuous. Alternatively, $f$ induces a homomorphism between the Lindebaum algebras of $M$ and $N$, which in turns induces, functorially, a morphism of type spaces.

Is there anything known about when such a morphism (of type spaces) is necessarily induced by an elementary extension? I'd be interested in any results along these lines in any setting.

If $M$ and $N$ are two $L$-structures, and $f: M \rightarrow N$ is an elementary extension, then given any subset $A$ of $M$, $f$ induces in a natural way a morphism $S^M_n(A) \rightarrow S^N_n(f(A))$ of type spaces which is in fact a homeomorphism of Stone spaces (in fact the underlying sets are equal: see the answer below).

What results exist about converse statements? If two type spaces are equal, when can you say it's because of the existence of an elementary embedding? What if you have a non-trivial homeomorphism, or some more general injective continuous map? Can you sometimes conclude that that it's induced by a suitable morphism between the structures?