A relative pseudomonad generalizes a pseudomonad relative to a 2-functor. This enables the construction of Kleisli bicategories of pseudomonads on $Prof$ (e.g. for defining generalized multicategories) by means of pseudo-distributive laws as if the presheaf category construction were a monad with Prof as its Kleisli bicategory, even though it is not for size reasons.

An example is the free cocompletion construction on a small category.

- Marcelo Fiore, Nicola Gambino, Martin Hyland, Glynn Winskel,
*Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures*, arxiv

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