It is well known that, given an endofunctor H on a category C ,
the initial (A+H-)-algebras (if existing), i.e. , the algebras
of (wellfounded) H-terms over different variable supplies A,
give rise to a monad with substitution as the extension operation
(the free monad induced by the functor H). Moss [17]
and Aczel, Adámek, Milius and Velebil [12] have shown
that a similar monad, which even enjoys the additional special
property of having iterations for all guarded substitution rules
(complete iterativeness), arises from the inverses of the final (A+H-)-coalgebras (if existing), i.e. , the algebras of
non-wellfounded H-terms. We show that, upon an appropriate
generalization of the notion of substitution, the same can more
generally be said about the initial T'(A,-)-algebras resp. the
inverses of the final T'(A,-)-coalgebras for any endobifunctor
T' on any category C such that the functors T'(-,X)
uniformly carry a monad structure.