Robert Dawson says "am I being naive?". Well, Robert, you asked for it, but yes you are, because you're assuming excluded middle. Pfn is the Kleisli category (of *free* algebras) for the partial element (lift) monad. Classically, every (Eilenberg-Moore) algebra is free, and so what Robert & Mike say is true. Intuitionistically, lift(1) is the object of truth values. Every algebra carries a partial order, which, for free algebras, is a powerset in each slice. In general it need not be distributive, because the category of algebras is closed under retracts (splitting idempotents). Indeed any algebra for a monad is a coequaliser of a pair of maps between free algebras, so colimits naturally take you outside the Kleisli category. The diagram may nevertheless still have a (different) colimit inside. This makes the question of colimits a little more complicated than Mike suggests, but then he's probably still the best person to say exactly how complete and cocomplete the category is. Here is an example of computer science interest, and a counterexample to what I claimed (privately) at the recent Manchester PSSL. David Rydeheard's book discusses unification (as used in logic and functional programming) as a coequaliser in the Kleisli category for the monad for an algebraic theory with no equations. A very simple example shows that this is *not* the coequaliser in the category of algebras. Take one unary operation f and two variables x,y. Then as a unification problem the equation f(x)=f(y) implies x=y, but the coequaliser as an algebra is essentially "N with two zeroes". Paul Taylor