  Coproducts of monads on Set have arisen in both the study of computational
effects and universal algebra.
  We describe coproducts of consistent monads on Set by an initial algebra
formula, and prove also the converse: if the coproduct exists, so do the
required initial algebras. That formula was, in the case of ideal monads, also
used by Ghani and Uustalu. We deduce that coproduct embeddings of consistent
monads are injective; and that a coproduct of injective monad morphisms is
injective.
  Two consistent monads have a coproduct iff either they have arbitrarily large
common fixpoints, or one is an exception monad, possibly modified to preserve
the empty set. Hence a consistent monad has a coproduct with every monad iff it
is an exception monad, possibly modified to preserve the empty set. We also
show other fixpoint results, including that a functor (not constant on nonempty
sets) is finitary iff every sufficiently large cardinal is a fixpoint.
