Generalizing substitution

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 [2] 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. Résumé. ... 1991 Mathematics Subject Classification. 08B20, 18C15, 18C50.