Arkham Mansion--Part 7

Scan the Warden's Office to Isolate a Forensic Trail

Scan the object on the ground to obtain his DNA trail.

Locate the Warden Using His DNA Trail

Use detective mode to follow the DNA trail back outside.

Proceed through the abandoned tunnel by grappling to the ledges above the closed gates until you're overlooking a group of baddies below. Two of them are armed, making this an ideal time to use you Multi-Batarang if you've unlocked it. You should be able to target the two armed inmates (they're the ones closest to you) and knock them out, giving you a chance to drop down and take them out before turning your attention to the remaining enemies. Once clear, exit through the large door into Arkham West.

Back in Arkham West, you'll find the DNA trail leads up a staircase to the west to a door guarded by several inmates. However, before venturing up there, you should grapple to the nearby sniper tower to take down the gunner there first. After clearing the area, proceed through the door leading into the Penitentiary.