Extensional characterization of non-deterministic finite state transductions
The best reference is probably the book Rational transductions and context-free languages by J. Berstel.

First recall that a function $f$ from $A^*$ to $B^*$ is realized by a transducer (= non-deterministic FST) if and only if it is rational, that is, if the set $\{(u, f(u)) \mid u \in A^* \}$ is a rational subset of $A^* \times B^*$.

You can find several other relevant characterizations of rational functions in Chapter 4 of Berstel's book.


(Eilenberg 1974. See Prop. 4.1 and Corollary 4.3) A function from $A^*$ to $B^*$ is rational if and only if it can be realized by a normalized unambiguous transducer.
(Schützenberger 1976. See theorem 4.5) A function is rational if and only if it admits a semimonomial representation.
(Elgot and Mezei 1965. See Theorem 5.2) A function is rational if and only if it is the composition of a left sequential transduction by a right sequential transducer.


See also A. Arnold and M. Latteux, A new proof of two theorems about rational transductions, Theoret. Comput. Sci. 2 (1979), 261–263.