Combining a monad and a comonad

We give previous termanext term systematic treatment of distributivity for previous termanext termprevious termmonadnext term and previous termanext termprevious termcomonadnext term as arises in giving category theoretic accounts of operational and denotational semantics, and in giving an intensional denotational semantics. We do this axiomatically, in terms of previous termanext termprevious termmonadnext term and previous termanext termprevious termcomonadnext term in previous termanext term 2-category, giving accounts of the Eilenberg–Moore and Kleisli constructions. We analyse the eight possible relationships, deducing that two pairs are isomorphic, but that the other pairs are all distinct. We develop those 2-categorical definitions necessary to support this analysis.