Every regular function is defined by a string-to-string mso transduction (Theorem C.4.8 of Transducers, Engelfriet–Hoogeboom, the implication from regular to transduction). A regular function is computed by a two-way transducer (Theorem C.2.9); the transduction's elements are the pairs of a configuration of its run and an index into the output produced there, and all its formulas — which configurations are reached, which letter is produced, which of two configurations comes first — are regular properties of the input with marked positions, hence mso-definable by Büchi's theorem.