??so is there some nice "monadic / anti-monadic factorization" of geometric morphisms??? ... ??and so forth ... ??...

???but ... ??what about something about stuff about ... "n-stage monadic functor" and so forth .... ???(??hmm, annoying terminology here... "binary operation" vs "secondary operation" ... ???...) .... ???something about ... "non-idempotent (??or something???) factorization system" ??? .... and so forth .... ???....