?? consistency check ... ?? take cofan cocone and toric quasicoherent sheaf, and "evaluate former at latter" ... ?? but then also ... ?? take the tqcs and express it in "graded" picture (?? in appropriate case ... ??? ....) ... ??? and take the cocone and .... ??? try expressing it as a "toric quasiherent co-sheaf" .... ???? and see whether the cosheaf-sheaf pairing here matches the original evaluation .... ???? .... ???? .....