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A functor gives an essential geometric morphism between toposes and , these are given by pre-composition, left, and right kan extensions. We have the same situation for V-enriched categories and and their V-enriched presheaf categories, taking V to be a cosmos. Is there a name for this construction, or is it written down somewhere?
Beyond "essential geometric morphism induced by ", I'm not personally aware of any particular name. :thinking:
Or perhaps not everyone calls them geometric morphisms in the enriched setting?