You start the game inside The Batcave. Keep moving forward without any delays, because you won't encounter any inmates in this section of the caves. Use the glide ability to make your way to a lower platform and after that take the stairs to reach Batman's computer. Watch a new cut-scene before moving on with the mission.
Notice that you've been equipped with a new gadget, called a Batclaw. Return to the lower platform and find a bridge with a large gap located in the middle. Choose the Batclaw from your inventory and grab one of the crates. Each of these objects needs to be pulled towards Batman by pressing the action key. Repeat this once or twice to clear a path and perform a jump towards a new section of the balcony.
Head forward and open a new door to make your way to the Old Sewer. You must choose the Batclaw from your inventory and use it to pull a grating located directly above you. Once this is done use your standard hook to reach the ventilation shaft. Head forward and exit the vents carefully when you reach the other end.
One of the inmates is standing to the left of your current position. Make sure that you're holding the crouch key. Approach him slowly and perform a silent takedown. Head straight and then turn right. More enemy units are stationed nearby and you'll have to attack them. Use gadgets if you want to and don't forget to defend yourself with a cape against opponents equipped with knives.
Proceed to the east and then turn north. Eventually you should be able to find a new door leading to the Main Sewer Junction. Head forward here and after that turn right. Approach the left wall and grab the ledge to climb up.
You'll have to perform a single jump here in order to land on a nearby metal balcony. Turn right and perform a second jump. You should be careful in this section of the caves, because it'll be easy to make a mistake. Head towards a brighter area, performing two other jumps along the way. It shouldn't be a problem as long as you're running in a straight line.
Once you've finished jumping you'll end up in the corner of the room. Turn left and use the first collapsed column to reach an upper platform. Turn left here and make a jump towards a new structure. From there perform a second jump, landing on a small concrete platform (screenshot #2).
Turn left for the last time. Use the second collapsed column to reach an upper ledge. Once you're at the top and perform the last jump in order to reach a nearby platform. Head on to your left and locate a door leading to Surface Access.
Turn right upon your arrival in the new area. Keep going straight until you'll be forced to use an Explosive Gel to blow up a small section of the wall. Don't forget to use detective mode to find this wall. Detonate the charge and jump over a small abyss. Opening the door will return you to Arkham North.