??so in the toric case maybe we have the idea that the toric quasicoherent sheaves are the coalgebras for a lex comonad on the toric non-quasicoherent sheaves ... ??so then what about non-toric analog?? ... ??maybe monoidal comonad or something ?? ... ???something about bit about... ??relationship between strong monoidalness and adjoints between lax monoidal functors ??? ... and so forth .... ???....