Type matchups are reversed. Unlike an inverse battle, immunities are reversed too and types that resist or are weak to themselves remain as so.
So under normal battle conditions, Bug is strong against Dark and Psychic (while both deal normal damage to Bug), Ghost is immune to (and deals normal damage to) Fighting while Fairy is immune to (and is strong against) Dragon. Bug would instead be weak to (and deals normal damage to) Dark and Psychic, Fighting would be immune (while dealing normal damage) to Ghost and Dragon would be immune to while also being strong against Fairy.