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For every finitary monad   $T$   on sets and every endofunctor   $F$   on the category of   $T$  -algebras, we introduce the concept of an ffg-Elgot algebra for   $F$  , i.e. an algebra admitting coherent solutions for finite systems of recursive equations with effects represented by the monad   $T$  . The goal is to study the existence and construction of free ffg-Elgot algebras. To this end, we investigate the locally ffg fixed point   $\varphi F$  , i.e. the colimit of all   $F$  -coalgebras with free finitely generated carrier, which is shown to be the initial ffg-Elgot algebra. This is the technical foundation for our main result: the category of ffg-Elgot algebras is monadic over the category of   $T$  -algebras.

On the behaviour of coalgebras with side effects and algebras with effectful iteration