Relative Pseudomonads, Kleisli Bicategories, and Substitution Monoidal Structures

We introduce the notion of a relative pseudomonad, which generalizes the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonads on the Kleisli bicategory of a relative pseudomonad. The results are applied to define several pseudomonads on the bicategory of profunctors in an homogeneous way, thus providing a uniform approach to the definition of bicategories that are of interest in operad theory, mathematical logic, and theoretical computer science.