?? are we really claiming that geometric morphisms between accidental toposes are surjective precisely in case they're essential ??? .... ?? and if so then why didn't we notice it before ... if we didn't .... ??? ....

?? maybe even for geometric morphisms merely into accidental toposes ??? ...

?? filteredly cocomplete picture here ??? ....

?? well, for merely into .... ??? couldn't you have a geometric morphism defined on a discrete sum, with one component taking care of the surjectiveness, and the other taking care of the non-essentialness ??? ..... hmmmmmm ...... ?????

?? well, what about merely out of ???? .... ?? for merely out of, essentialness clearly doesn't imply surjective ??? ... ?? from looking at affine case ???

?? while for merely into, surjective pretty clearly doesn't imply essential .... ????

?? for merely into, "essential implies surjective" seems plausible at the moment .... ???? ....

?? for merely out of, how plausible is "surjective implies essential" ???? ?? not terribly, at the moment ??? .....

?? surjective as "anti-injective" by factorization theorem ??? ....

?? hmmm, so how does "essential is equivalent to surjective" stand up in case where domain and co-domain are both possibly non-affine accidental toposes ?? ... ???? ...

??? consider inclusion of "plane minus axis" into punctured plane .... ??? ....

?? hmmm ... point included into line .... torus included into line .... ??? both surjective ??? .... ???? .... ???? .....

n to 1 .... "restricting" along this as "inclusion of trivial actions" .... left adjoint to that as .... ?? taking orbits ??? ....

?? coalgebra for comonad here .... ??? right adjoint then left .... ????