?? some peculiar stuff going on here ... ?? inclusion order structure on subtoposes ... ?? in particular, on (certain ...) 1-morphisms with varying domain but fixed co-domain ... ?? somehow becoming (?? ...) order on morphisms in single hom-space, associated with 2-morphisms .... ??? maybe factorization system where some part is systematically endo ? ... ??? well, maybe already have something ilke that, with line bundles (?? ...), but then also with localizations ?? ... ??? ...

?? "relatively invertible" ... ?? relative inverse ?? ... ?? ...

?? anyway, take another stab here at giving nice explicit description of geometric morphisms and 2-morphisms between accidental toposes .... before trying to test/prove guess ... ?? ...

?? so let x and y be toric varieties ... ?? then we want to describe category of geometric morphisms from accidental topos of x to that of y .... ?? ...

?? so ... object as .... ??? .... ?? dense toric open o of y, together with invertible tqcs i over o, together with "inverse-affine-preserving" toric map m from o to x ....

??? morphism from (o,i,m) to (o',i',m') as ... ?? inclusion of o in o', with m = m' restricted to o, and a morphism f from i to the pullback of i' to o ...

?? maybe hopeless to straigthen out whether f should go that way or the other, on the grounds that usinf the inverse line bundles would switch the convention .... ??? ....