If your car is coated, then the detailer polished the area, the coating is firstly gone in that area, so rightfully they need to reapply the coating in that area.

Secondly, how deep do u think is the scratch? has it gone through that u can see the base primer colour (usually grey)? You can try to assess the depth of the scratch by using your fingernail and lightly "scratch" across the scratch. If it feels very deep that u may need to touchup, otherwise u can try to compound it a few rounds by hand. U can use a microfiber towel and apply a dash of compound then rub on it.

Again, please remember that after u have done coating, any polishing/compounding will most likely remove the coating. So it should be reapplied.

Hmm, don't think touch up would do much cuz u be just touching up on top of the coating. As the coating wears, the touch up will disappear and the scratch will come back. Perhaps it be best to bring back to detailer to redo that section, else wait a year or two after the coating has worn of to re-detail the car again.