Any geometric morphism between localic topoi is localic.
Any geometric embedding is localic.
Thus, in general we regard a localic geometric morphism as exhibiting as a “localic -topos”.
This is supported by the following fact.
For any base , the 2-category of localic -toposes (i.e. the full sub-2-category
Concretely, the internal locale in defined by a localic geometric morphism is the formal dual to the direct image of the subobject classifier of , regarded as an internal poset (as described there) and is equivalent to the internal category of sheaves over .
The last bit is lemma 1.2 in (Johnstone).
Localic geometric morphisms are the right class of a 2-categorical orthogonal factorization system on the 2-category Topos of topoi. The corresponding left class is the class of hyperconnected geometric morphisms.
This is the main statement in (Johnstone).
Localic geometric morphisms are defined in def. 4.6.1 of
The discussion there is based on