??so let's consider the "separated coherent presheafs over p^1" ...(not very systematic nonce terminology here...)

??or should we maybe even try p^0 instead? ...

let's take the cokernel of the mate of the obvious epi...

hmmm... ??what about the idea that if (for example...) we're looking for objects in [the theory of a line object equipped with an epi from 1] that must become adjoint to the cokernel of the mate of the epi, one possibility to check might be the internal hom [cokernel of mate of the epi,1] ?? ... ??or something?? ... on the other hand there's also the adjoint in the quasicoherent sheaf category, included back via fully faithful right adjoint ... ???....

??the obvious epi is the change-of-filtration morphism... 1 = "all born at 0", L = "all born at -1" ... ?? ...

??then the mate of this is the change-of-filtration from "all born at 1" to "all born at 0" ?? .... ??so then what's the cokernel of this?? ??hmm, shouldn't it be 0???? .... because the mate is again epi .... ???? ...

???doesn't it seem like something's screwed up here???.....

??like i'm almost claiming that there's an "equation" that holds in the free example but that doens't occur in some other example ... ???or something?? ...

???something about ... ??"mate for life" ??? ...??...

oh wait a minute... this is the p^0 example... ??maybe the mate really is epi in this example ... ????... ??for life?? ... ??yes, i think so...


??is that going to screw up the purpose of the example, so we might have to use p^1 instead of p^0 ?? or what??