??so the (??co-)weighted limit of a set-valued functor f on s wrt a (??co-)weight w is essentially just hom_[_set_^s](w,f) ... ??...

??and then when f is x-valued the above holds "yoneda-wise" ??...

??and if we take x = _set_^op then we should get the hopefully pretty familiar concept of "weighted co-limit" ??... or something?? ???where a "weight" is (??now...) a contravariant set-valued functor on the diagram scheme ... ??or something??

??so let's try for example the case where the diagram scheme s looks like "m:d->c"... so a diagram f can be thought of as a "family of sets", with f(c) being the family and the fibers of f(m) being the sets in it... and take the co-weight w to be a "singleton family of doubletons"...

so the co-weighted limit of a family of sets in this case is ess just "the sum of the squares" (of the fibers)... or something...

??whereas the weighted _co-_limit of a family of sets wrt the same w interpreted now as a weight rather than co-weight (i'm being sloppy here about the way that the diagram scheme s is self-opposite here...) is ... ??...something like "one point for each occupied set and two points for each unoccupied one" ??? ... that seems a bit awkward... ??does it really work, yoneda-wise?? well, i tried working it out in my _real_ notebook... the handwritten one, which at the moment is windows journal file notebook358 on my tablet/laptop... and it seemed to work out reasonably well, though it didn't completely get rid of the awkwardness... i tried using 2 as my "point representer" in _set_^op, and it worked out to something like... "given a family d of sets, if i paint a subfamily of them blue and another subfamily of them green, such that the sum of the blue subfamily is the same as the sum of the green subfamily (as a subset of the sum of the whole family), then it's equivalent to selecting a subset of [one point for each occupied set in the family, and two (blue and green) for each unoccupied one]" (or something... ??in this story it's ok if something is painted both blue and green?? ...) ... ??because unoccupied sets don't contribute to the union of the family, so blue and green can go their separate ways in that case... there really should be some less awkward intuition about this though... ??maybe some intuitive approach towards "tensor products" that i've forgotten or never learned... or something...

i suppose that you could say something like "take the weighted sum of the underlying discrete diagram, and consider it as the vertexes of a graph where the edges ride the morphisms of the diagram scheme... (??or something?? ??something about "riding in both directions"... once covariantly (for the diagram) and once contravariantly (for the weight)... ??or something??...)... and then take the components of that graph" ... or something like that... ??...

perhaps that _is_ an intuition about set-valued tensor products that i've forgotten or never quite learned... ??...

??anyway, i guess that the primary lesson that i'm supposed to be learning from this at the moment is just about thinking of (co-)weighted limits as hom-objects (or something... ??so co-weight and diagram have same variance, so you can hom co-weight into diagram... ??or something...) and weighted colimits as tensor products (or something... so weight and diagram have opposite variance... like "contracting upper index with lower" or something ... einstein convention or something...)...

(??by the way, what about "hom : parallel :: tensor product : serial" here?? or something?? ... and so forth ... ??...)