The Oracle's Chamber
After the cutscene, run along the crumbling pathway, trying to catch up to the old-now-young dude. Watch out for green grapple points, as they will be very important.
Keep running and jumping until you get to a slide section. Now, be prepared for multiple QTEs where you are only hitting X. After the quick cutscene, time to kill this guy. Oh boy, is this guy tough. He employs lots of tactics. Any time you see a green circle on the ground, that is a slow-time attack. You will act much slower while in a circle. Any time you see small rocks on the edge of the arena, grapple them to hurl at the guy.
He will also cause the arena to fall apart and come back together, so make sure you are always on solid ground. Sometimes, he will jump up on the pillars around the arena. When this happens, smash the pillars so that he falls to the ground. Most of the way through the battle, he will retreat to the very back, and completely encircle himself in green magic.
When this happens, run all the way to the front (towards the screen) so that you don't fall down the chasm. Towards the end of the fight, you will have the opportunity to grapple with him. Make sure your QTE fingers are ready to fly.
At the end of the fight, you will unlock the Quaid trophy as well as the Amulet of Uroborus.