The fight to take over Belltown district (District Showdown) will be possible only after D.U.P. forces control less than 30% of a district. Go to the place marked with a phone icon (large billboard on the rooftop) and make a call.
I suggest starting this battle with eliminating weaker D.U.P. soldiers who are occupying two rooftops (one building is a carwash). Once you deal with them you'll be able to focus on main targets - two elite soldiers. Try to split those two and then attack them one after other because it will take some time to kill them. Do not forget about your stronger attack and withdraw when you get into troubles - simultaneous attack from two sides can quickly finish you.
Rewards for taking over Belltown district are:
- Unlocking fast travel
- Getting rid of all D.U.P. soldiers in the area
- Unlocking Delsin's unique clothes - Lone Wolf