When you reach Wall Market in Chapter 9 you'll be able to make a variety of choices that will determine which side quest you unlock later in the chapter (after parting ways with Aerith after the Undergound Colosseum). To increase the odds of unlocking this quest, do the following in Wall Market:
When speaking to Chocobo Sam and describing Tifa, select “She’s great at handling the books”.
Talk to the man outside the hotel and respond with "How much?".
When you find Johnny, follow him to the Honeybee Inn and talk to him. Respond with "No", or just ignore the event altogether.
When Sam does the coin flip, select “No deal".
When forced to pick a massage course while talking to Madam M, opt for the "Luxury Course - 3,0000 G".
When Aerith asks how you like her outfit, select “It matter what I think?”.
If you follow this sequence, you should receive this Quest from the Colosseum Gatekeeper when returning to Madam M after completing both Burning Thighs and The Price of Thievery. It looks like the Gatekeeper was so impressed with your skills earlier that he has one more special match for you to undertake. Before you head there however, be sure to set Elemental-Lightning on Cloud's weapon, it'll make your life a lot easier. Once you're ready, make your way back down to the Colosseum and select the "Special Match" to get started.
|Jury-Rigged Cutter||Lightning||Turbo Ether|
You'll be up alone against a new type of Sweeper, the Jury-Rigged Cutter and this one is deadlier than its counterparts. As is true of all mechanical enemies in Final Fantasy VII Remake, its weakness is Lightning so having it set to Cloud's weapon will allow you to strike at the weakness with both Physical attacks while also tossing casts of Thunder(ra) to drive up it's Stagger gauge.
Don't be too careless however, it can deal hefty damage to you if you're not careful, especially with Buzzsaw Blitz if you're caught in the middle of it. When you see the name above it's head, dodge around to the side and get some free attacks on it. Once again, once it's Staggered, Infinity's End will make short work of it. Return to Madam M following your victory to obtain an Arcane Scepter, a new weapon for Aerith.