Initial algebras and terminal coalgebras in many-sorted sets

For endofunctors F of many-sorted sets the iterative construction of initial algebras is proved to converge whenever F has an initial algebra. In case of one-sorted sets the convergence takes n steps where either n is an infinite regular cardinal or it is at most 3. Dually, the existence of a terminal coalgebra implies that the iterative construction of a terminal coalgebra converges. As demonstrated by James Worell the number of steps here need not to be a cardinal even in case of one sort : it is ω + ω for the finite power-set functor. For related categories e.g. graphs the above results do not hold: we present non-constructive initial algebras and terminal coalgebras.