?? "inverse image of affine is affine" ... ??? geometric / algebraic / conceptual aspects .... ??? .... ??? toric vs non-toric case .... ??? ....

?? ... accidental injective geometric morphism ...

?? ... accidental surjective geometric morphism ... ?? lex comonad ... coalgebras .... ?? ...

?? non-affine quasi-affine ... ??? is it true that this gives injective geometric morphism for "domain-narrowing", but not surjective geometric morphism as toric map ??

?? trying to straighten out blame-apportionment ... lex vs left adjoint ... ?? also injective and surjective .... ??? .... ?? toric and non-toric ... ??? ... ??? ... ?? being a (partially-defined) map vs some more general sort of "correspondence" ... ?? ....

?? forcing pullback condition to hold, vs inclusion of those actions satisfying the condition ... ???? ...