There are various models of algebras in which certain recursive equations are uniquely solvable. A simple one was introduced by Capretta, Uustalu and Vene: given an endofunctor H, recursive equations are formalized as coalgebras for H. An algebra for H is called corecursive if every coalgebra has a unique 'solution', i.e., a unique coalgebra-to-algebra morphism.
It turns out that this condition is too weak to capture 'interesting' recursion. Milius defined completely iterative algebras (cia's, for short) as algebras on an object A such that every coalgebra for H(-)+A has a unique solution. This makes the range of recursion much wider. Not surprisingly, for most of 'everyday' functors corecursive algebras are not cia's, in general.
Recently Milius and the author proved that if H preserves countable coproducts, then for every functor H(-)+Y corecusive algebras are precisely the cia's. This may sound as a too special result, since preservation of coproducts is rare. However, we also proved that if H is a finitary set functor, then it has the above form whenever cia's coincide with corecursive algebras.