??so wa barr-beck (or something ...) distributivity decompositions of yoneda monad (or something ... and so forth ...) and something about ... "filteredly cocomplete analog of connected projective" ??.... and so forth ...

??examples... ??...

??connected ...

??projective ...

??"compact" or something ??? ....

and so forth ... ???....

??something about "relative cocompletions" of various kinds, vs ... ??other ideas, or something ???...

??something about ... ??first adjoining connected colimits, then discrete ?? ... and so forth ... ??....

??so what about for example... ??freely adjoining filtered colimits to _finset_ ... ??vs relatively (??to finite colimits, or something ?? ...) freely adjoining all small colimits to it ... ??? .... and so forth ... ???...