?? cocones (or co-yoneda morphisms ...) of terminal type in colimits sketch ??? ... ???seems weird??? ... but ok ?? .... "invertiblization" ... ???....

??topos case ??? .... ????

?? ... example... ?? "product span" ... ???

c :=

"source" -> "arrow" <- "target" ??walking arrow vs walking source-target-pair ... ??? ??co-yoneda morphism .... ??? walking source-target-pair -> walking arrow

functor f : c^op -> _set_ .... "span" ...

co-yoneda morphism in c goes to co-yoneda morphism in _set_^op under f ...

actual sources X actual targets <- actual arrows ... ???.... ??? some confusion here ... ??formal vs actual in _set_^op ??? .... ?? "c respects given co-yoneda morphism" ??? .... ????..... ??co-yoneda morphism coming from a cocone ... ????.... ???example ??? ???on _set_^op here ?? ... "[actual arrows,-]" <- "[actual sources,-]" + "[actual targets,-]" ???confusion here .... try again ... co-span y -> x <- z in category c ... ?? push particular special co-yoneda morphism forward from walking co-span to c ... special one = walking y + walking z -> walking x ???.....

"[-,y]" + "[-,z]" -> "[-,x]" ...??yes, seems right ?? ....

pushing "[-,y]" (for example) forward ...

??say for example along "yz,x,-" ... ???

???well, pushforward of representable is representable, right ??? ...

???so maybe now we're ready to pretend that x,y,z live in _set_^op ... ??? ??thus forming a span / bipartite graph ... ????..

"functions on sources" + "functions on targets" -> "functions on arrows" ....

?? "you can get a function on arrows either from a function on sources, or from a function on targets" ... "function on arrows that only depends on their sources, vs one that only depends on their targets" .... ??plenty of _other_ functions on arrows though ... ??? even if it's a complete bipartite graph ???? ..... ???confusion again ???.....

???_the_ b-cobased cocone in walking b-cobased cocone as a colimit cocone .... (??why???? ...) ????but the corresponding co-yoneda morphism doesn't seem to be invertible ?????? ..... ????confusion ??? .... but so sharp that we ought to be able to resolve it ... ????...

(????general colimits in walking b-cobased cocone ????)

??let's consider as analog of "cocone with given cobase", "co-yoneda morphism with given (presheaf) domain" ... ??but then .... ???maybe the key thing that we've been screwing up is in fact just that ... ??? "universality" of a co-yoneda morphism is not equivalent to invertibility ??? .... ??but then what _is_ it like ???...

??? "a morphism from the co-domain to another actual object t is essentially the same thing as a co-yoneda morphism from the (virtual) domain to t" .... ???....

??? "representablization" and "total category" ??? .... ...??that's what it's like ??? ..... ????....

(there _are_ invertible co-yoneda morphisms, but those aren't merely universal .... ???they come from co-bases which have a terminal object .... ????..... diagram scheme where all you have to do is evaluate at the terminal node to get the colimit ... ???.... ??? "vacuous colimit" in some sense ??? ....)

???so what are we saying??? .... something like .... ???corresponding to a reflective subcategory (??to begin with, let's say of a presheaf category ... ??? .. well, hmm, actually this is causing me some confusion ... blurring distinction between "grothendieck" and "lawvere-tierney" viewpoints ... which is why all those ?'s here... but i'll try to clear it up ...) is, for each object in the site................??????????????? ??? "promoted to true" ... categorified true ... ???kind of thing that people probably do categorify a lot ???

???there _is_ some sort of "image/kernel duality" (??...???...) here, like i think i was vaguely feeling .... ???? "dense vs closed" .... ???? .... ???? .....