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Thanks, that seems right.
I have made the typesetting of the factorization come out with the fully faithful functors displaying as hooked arrows. Also I have added mentioning that those full subcategories which are equivalent are he “$E$”, from item 11.
But something else is still wrong: the entry claims that an adjunction is idempotent when its induced monad and comonad is. This contradicts the statement at idempotent monad, which says that this is true (only) if one of the two adjoints is fully faithful.
Oh, sure, right. Thanks.
I have added this original reference for the characterization of idempotent adjunctions stated here:
Also I have adjusted the formatting and cross-linking of the only reference which used to be offered (now here).
also pointer to:
added pointer to:
I have given the list of examples more formatting and more hyperlinks.
Then I have added the example
$TopSp \underoverset{ \underset{ Cdfflg }{\longrightarrow} }{ \overset{ Dtplg }{\longleftarrow} }{\phantom{AA}\bot\phantom{AA}} DifflgSp$(by !include
-ing it).
The earliest textbook reference for the full characterization of idempotent adjunctions that I have found so far is still Grandis 2021. Is there an earlier textbook that states the Proposition in citable form?
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