The start point of this quest is located on the building's rooftop in the northern part of the Market District and before starting it you can be asked to perform some additional activities (if it is like this, you should check information in the City chapter).
After starting the quest jump off the rooftop and approach of people protesting against bio-terrorist. Now you will have opportunity to get Bad Karma. If you follow hero's path then ignore them but if you follow villain's path then murder all protesters. Regardless of your behavior listen to the conversation with Reggie and get to the rooftop of the Olaf's Sea Shack building located south from here. Now you have to approach cartons and press touchpad to take cover.
Wait for a cutscene presenting Fetch. Delsin will absorb a basic power (Neon) of the girl and she will start to flee. Try out the new power. Press and hold circle button to activate Light Speed which allows you to move faster. Unfortunately this power works only for few seconds and when you slow down you have to hold the circle button again.
Use Light Speed to follow Fetch (purple trace) and do not worry if you slightly go off the route because you'll be re-routed in a subtle way. During the chase you'll find out that Light Speed can be used also to run over walls in order to get to rooftops. Use it wisely and no not lose Fetch. Keep chasing her.
Once you get to the theater rooftop the chase will be over and the game will show you a new cutscene. After that you'll have to face Fetch (she is one of game bosses). At the beginning try to familiarize yourself with her two standard attacks. She will try to deal you damage with a laser and you can avoid it by using Light Speed and by hiding behind covers.
Do not try to approach Fetch because she could then use energy explosion and such attack can strongly weaken or even kill you (if you are already wounded). It is worth mentioning that you do not have to fight with her in a melee to win this duel, so keep your distance from her.
The fight with Fetch is divided into two parts. In the first one you have to absorb energy from nine neons located in the building (this is also a way to refill Neon energy). Each time you approach neon position yourself in such place that Fetch can't hit you and then start absorbing energy (press touchpad).
Clear out the entire ground floor, finding all neons and absorbing their energy. But this is not all, because there are some neons on upper balcony. To reach them run over walls, watching out for Fetch. When you absorb energy from all neons, the second part of the fight begins.
In the second part of fight you'll be able to use basic distant attack of Neon power. It is Neon Beam which can be activated by R2 button (it is equivalent of Smoke Shot from smoke attacks category). Start using this attack to weaken Fetch but be aware that she will also use it, so keep moving and/or use covers. The girl can hide from time to time and in such situations track her down by following her traces.
You need to keep fighting, until you deprive Fetch of her entire health bar, which will start another cutscene. Towards the end of that cutscene, you will have to make another important choice (the above screenshot). If you want to gain good karma you need to promise Reggie, that you will bring Fetch back to good path (the blue option - Redeem Fetch). If you want to gain bad karma, you need to select the variant that allows you to train Fetch to be a ruthless killer (the red option - Corrupt Fetch).
Regardless of your decision this quest will be over and you'll unlock the next one - Light It Up.