On Rational Monads and Free Iterative Theories

For every finitary endofunctor H of Set a rational algebraic theory (or a rational finitary monad) R is defined by means of solving all finitary flat systems of recursive equations over H. This generalizes the result of Elgot and his coauthors, describing a free iterative theory of a polynomial endofunctor H as the theory R of all rational infinite trees. We present a coalgebraic proof that R is a free iterative theory on H for every finitary endofunctor H, which is substantially simpler than the previous proof by Elgot et al., as well as our previous proof. This result holds for more general categories than Set.