Parametrized iterativity of an algebra means the existence of unique solutions of all finitary recursive systems of equations where recursion is allowed to use only some variables (chosen as a parameter). We show how such algebras can be introduced in an arbitrary category A by employing a base, i.e., an operation interpreting objects of A as monads on A. For every base we prove that free base algebras and free iterative base algebras exist. The main result is a coalgebraic construction of the

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... atter: all equation morphisms form a diagram whose colimit is proved to be a free iterative base algebra. (Milius). 1 Finitary functors are those preserving filtered colimits. Finitary monads are monads with a finitary underlying functor. 0890-5401/$ -see front matter