Multi-monadic categories on $Set$ can be characterized in the following way: they are regular, with connected limits, with coequalizers of coequalizable pairs, their equivalence relations are effective, their forgetful functors preserve coequalizers of equivalence relations and reflect isomorphisms. Unlike monadic categories they need not have products. Examples include local rings, fields, inner spaces, locally compact spaces, locally compact groups, and complete ordered sets. (Diers 80, p.153)