between accidental toposes ... ?? especially non-affine ?? ...

?? "symmetric semi-monoidal geometric morphism" ... ?? geometricness requirement as not allowing for "cohomological" / "(relatively ...) non-affine" case ... ??? ... ?? absolute vs relative case here ??? .... ??? in absolute case topos manages to keep toposness, whereas in relative case geometric morphism loses its geometricness ??? .... ??? confuision ... level slip ... ?? ....

?? "symmetric semi-monoidal essential geometric morphism" ... ??

?? "essential symmetric semi-monoidal geometric morphism" ... ??

?? "symmetric semi-monoidal essential morphism" ... ?? maybe "correct" ?? ...

?? ambiguities / glitches in parsing some of these terms ?? ... (?? "nullary ambiguity" ?? ...)

?? further complications connected with "semi-" here ?? ...