something about ... ??weak dual of the generic object x (wrt the ag doctrine...) as 0 because of lack of any interesting x tensor ? -> 1 ... ???or something?? ... that is, something about... "too little raher than too co-much" ?? ... or something ... ??...

??so have we got _any_ examples where... ???"freely adjoining an adjoint for something causes something else to become that adjoint" ??? or something??

??something about... possible analog... noncommutative monoid... walking one-sided inverse situation ... ???forcing two-sided inverse to exist... ???or something?? ...

???so what _about_ P^1 case??? ....

??theory of epi e : 1+1 -> 1 ... with dual of cokernel of its mate ... st ... e is its mate's cokernel's mate's cokernel????

??punctured plane?? ??as "quasi-affine" ... ???...

??so ... ??consider prop as follows : ... ???...

?? e1,e2 : 1 -> 1 ...

"unit" ... 1 -> ( cok( 1 -> 1+1 ) tensor x ) ... ???maybe problematic??

"co-unit" ... ( cok( 1 -> 1+1 ) tensor x ) -> 1 ...???

?? x+x -> 1 ... ??st x -> x+x -> 1 is zero ... ???or something???


?? cok( 1 -> 1+1 ) tensor x ... ??as cok( x -> x+x ) ... ???

1 -> cok( x -> x+x )

cok( x -> x+x ) -> 1

???maybe we really need the prop for a dual pair of objects, and then... ??apply gabriel-ulmer duality ... ???or something???....

well, so... ???suppose that we have a "quadrant" of modules of k[x,y] ...

?? equipped with "cap" and "cup" operators ... ??...

?? and "permutation" operators ... ???...

???satisfying some hopefully obvious compatibility laws, and so forth... ??...

???and then some additional operators... ??relating to one of the objects being the cokernel object of "(x,y)" : 1 -> 1+1, and ... ??something about mate's cokernel's mate's cokernel ... ???...