From where the last fight ended, head up the grapple point to the ledge above.
Item Upgrade: Use the nearby grappling point to climb up and double jump across to the pile.
A passage will lead you to a larger area, studded with pillars.
Diary: A corpse will be leaning against a pillar to the left
Drop down and head along the linear path to the massive horn.
Shortly afterward you are back in the present, in the elevator with your companion heading for the courtyard out the front of the building. Head out the front and then take a left. Drop down the hole inside and then drop down again. Use the mirror of fate panel under your feet to reassemble the mirror. Complete the simple minigame to open the large door nearby. In the next room think of it as extreme version of a flying fox or ziplining. Grab onto the hook and carefully move forwards, dodging the flames.
Oh that's just great. Once you are across, a convienient bridge appears to let your companion across. Head into the new area to find a painful throwing range. You have to activate platforms by standing on the skull icon (makes them appear) and then throwing a blood dagger at them to make them flatten for your companion to jump on. Sounds easy right. Wait until you have to contend with the fireballs. If a fireball touches your companion it will instantly fail you.The fireballs are always aimed at you, instead of being in a pattern. So this sequence is a royal pain. Depending on your skill and luck, you may get through this with only light insanity.
Drop down to the courtyard and watch your next lot of enemies appear (who didn't see that coming?).