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Tom Hirschowitz (Apr 07 2021 at 09:58):In good cases, there is an equivalence between, say, finitary monad morphisms on a category and finitary monadic functors over . Does this correspondence have a name?
And second, more important question, how does it carry over to the heterogeneous setting, where may vary?
Specifically, if we restrict to sufficiently nice categories (e.g., locally finitely presentable) and monads (e.g., finitary), we can consider monad morphisms in Street's sense (in the formal theory of monads). They do yield morphisms between the induced monadic functors in the arrow category of . Is this again an equivalence? Does anyone have a reference for related facts?
Tom Hirschowitz (Apr 07 2021 at 10:41):I think I just checked that the monads / monadic functors works just the same in the heterogeneous case, except for functoriality in the monadic functors monads direction.
Tom Hirschowitz (Apr 07 2021 at 11:06):... which appears to also work, so I guess I'm just looking for confirmation that this is well known, and references if they exist.