We introduce a generalization of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed λ-calculus syntax and indexed containers. We show that the Kleisli and Eilenberg-Moore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions, relative monads are monoids in the functor category concerned and extend to monads, giving rise to a coreflection between relative monads and monads. Arrows are also an instance of relative monads.
Altenkirch, T., Chapman, J., & Uustalu, T. (2015). Monads need not be endofunctors. Logical Methods in Computer Science, 11(1), 1-40. https://doi.org/10.2168/LMCS-11%281%3A3%292015