Recognisable transducers over monads and comonads

In 2015, Bojańczyk introduced a class of recognizable languages over monads, which is a common generalization of regular languages for many different types of objects, including words, infinite words, and trees. In this talk, I am going to extend this definition to transducers. As it turns out, the natural construction requires the underlying functor to be both a monad and comonad. In addition to defining and discussing this new definition of recognizable transducers over monads and comonads, I am going to provide a few examples of such transducers and show that (under some natural conditions) they are closed under compositions.