Vetala is just at the bottom of the stairs. See here for details. Your reward is a piece of Cobaltite . A Librascope can also be found here.

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5x Pulsework Gladiators location (left) and Hope get’s wrecked just for fun (right).

After you’re farmed to your heart’s content, travel down the corridor to the next area where another Menhirrim stands watch and a Tyrant lurks. Kill it first and then talk to the Menhirrim to receive mark Mission 26. Proceed to the outer ring and approach another Mehirrim for a cutscene where that pesky fal’Cie blocks your path. Return to the sixth floor by heading back up the staircase (near the Pulsework Gladiator battle).