In a double battle, a Pokemon uses role play and aquires protean. Then, it uses entraintment on another Pokemon. Finally, that 3rd Pokemon with protean now, uses flying press. What type will it become?
It would be fighting type. The move is a fighting type move, it just has the effect of both Fighting and Flying. Here is it's description:
Flying Press deals damage. It is a Fighting type move, but deals a combination of Fighting and Flying type damage.
Hope this answers your question!
Source
Despite being a Fighting-type move, the damage dealt is actually a combination of Fighting and Flying types, and thus its effectiveness against a given Pokémon differs from other Fighting-type moves. -Bulbapedia
In a way, you could say that moves now have Primary typings. Flying Press' primary typing is Fighting, so the user would most likely become a Fighting type
