We generalise the notion of a distributive law between a monad and a comonad to consider weakened structures such as pointed or co-pointed endofunctors, or endofunctors. We investigate Eilenberg-Moore and Kleisli constructions for each of these possibilities. Then we consider two applications of these weakened notions of distributivity in detail. We characterise Turi and Plotkin's model of GSOS as a distributive law of a monad over a co-pointed endofunctor, and we analyse generalised coiteration and coalgebraic coinduction “up-to” in terms of a distributive law of the underlying pointed endofunctor of a monad over an endofunctor.
Lenisa, M., Power, J., & Watanabe, H. (2000). Distributivity for endofunctors, pointed and co-pointed endofunctors, monads and comonads. Electronic Notes in Theoretical Computer Science, 33, 230-260. https://doi.org/10.1016/S1571-0661(05)80350-0