so let's try looking at the flat toric quasicoherent sheaves over the projective line... in various pictures ... ??...

??_is_ this reasonably unambigous ??? .... ????.....

??we're expecting that the 3 possibilities are "NZZ", "ZZZ", and "ZZN" ?? ...???... ??does that fit with the graded action picture?? ... ???....

??confusion here about (??among other things??...) mapping out of a limit .... ???....

??what about something about ... ???whether tensor product of quasicoherent sheaves (and/or toric analog) can be thought of as "corrected" (?sheafified?? ... ??) tensor product of some more "pre-sheaf-y" underlying things... ???and whether that might be somehow relevant here... ??something about maybe getting left exact left adjoint by using _un_-corrected tensor product ... ??? ....

??anyway, the three left exact left adjoints should be given by ... tensoring the first coordinate with N, or either coordinate with Z, or the second coordinate with N ... ??? ... ????....