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Nov 10, 2025 at 16:37 comment added fosco I'm not turning this comment into an answer, mostly because it is an unsatisfactory answer (I would say it doesn't satisfy your request (B)). But one could rightly argue that monads have by now a long history outside of Category Theory, initiated by Moggi, as models for effects, while comonads model something like "behaviour" in their coKleisli category, see e.g. Jacobs' book on coalgebras (admittedly, stretching the analogy from endofunctors to comonads). There you have it: the "effect category" of a monad T and the "behaviour category" of a comonad S.
Nov 8, 2025 at 22:35 comment added varkor I welcome all the people who are anonymously downvoting the question to point out in what ways the question is badly formulated, or not according to the MathOverflow guidelines.
Nov 8, 2025 at 16:12 comment added Denis T Category of operations/category of free terms? (With justification that composition of morphisms in Kleisli category tautologically corresponds to substitution = multiplication in the monad.)
Nov 8, 2025 at 11:06 history asked varkor CC BY-SA 4.0