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A dedicated discussion of the comparison maps between any adjunction and its initial Kleisli- and terminal monadic adjunction is being alluded to in various related entries, but none of them has really been admitting to details or giving any concrete citations.
This entry is meant to fill that gap. It’s unfortunate that this important concept does not have a more descriptive name. I have added some words of disambiguation in order to account for this.
Best to give a citation (if not a proof).
I added another link, which hopefully gives enough details to act as a proof sketch.
Hopefully? :-)
Just to clarify, I am not asking for myself but for the nLab — I am asking for, say, when in ten years from now a kid happens upon this statement and starts a goose chase on MathOverflow asking random people about how to prove what they will call another one of those unjustified claims on the nLab.
Best to pre-empt this right when adding the claims.
Is there a name in the literature for a right adjoint functor whose comparison functor has a left adjoint? (Somewhat analogous to descent type, which specifies a different property of the comparison functor.)
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