Every regular function is computed by a streaming string transducer (Theorem C.3.2 of Transducers, the implication from regular to sst). Streaming string transducers are closed under post-composition with every prime regular function — a rational function through the Krohn–Rhodes decomposition of its Mealy machines, map reverse and map duplicate by keeping the register contents as tuples indexed by the separators inside them — and the identity is an sst.