?? grade-wise dual of quasicoherent sheaf over p^1 ?? ... ?? as quasiherent ?? ... ?? toric vs non-toric confusion here ?? ... ??? ....

?? or ... ??? grade-wise pre-dual .... ????? .....

?? or does it maybe work equally well both ways ?? .... ????? .... ??? ...

?? seems like the naive "graded" picture of a quasiherent cosheaf has a better chance of working than the naive "glued" picture ??? .... ??? because homming into a set takes colimits to limits .... ????? ....... ???? ..... ?? in graded picture there's colimit structure ??? .... ??? not completely clear what kind of structure there is in glued picture .... ??? .....

?? nice description of cosheafification here ??? ..... ???? ...

?? so consider models of accidental topos in graded picture ... ?? ... look at right adjoint parts ?? ... ?? and try to guess corresponding quasiherent co-sheaves ?? ... ?? ...