Promonad
References.
- Distributors at Work (Benabou, 2000)
- Coend Calculus (Loregian)
- Promonads and String Diagrams for Effectful Categories (Roman, 2022), Definition 3.7 takes this style of definition. Most sources will more succintly say it is a monoid in the bicategory of profunctors.
Topic.