Final Fantasy VII Remake
Corneo Colosseum

Once you have defeated Hell House in the main story of "The Town That Never Sleeps", the colosseum will open up to a series of challenges. And these are available in Chapter 9, and again in Chapter 14. Aerith can only do her battles in Chapter 9, while Barret and Tifa cannot compete in theirs until Chapter 14. You definitely want to do all of Cloud and Aerith's immediately after it becomes available Once you go past Aerith going to dress up with Madam M, you will lose all of her battles for good. Aerith has two battles she needs to complete:
Aerith Vs. Wild Animals
Aerith Vs. Shinra Thugs.
If you do not get these, you will either have to begin from a previous save file, or wait until you get Chapter Select after beating the game. It is actually to your benefit to complete Cloud and Aerith's single battles right after Hell House. Not only do you get their second tier limit breaks, you get an extra 10 SP from the manuscripts.

Each challange will pit you against 5 consecutive fights with a wide variety of enemies. There is no use of Items allowed, but you will gain 50% HP/MP at the beginning of each battle. Each of the battles have the same reward and enemies for each of the games difficulty levels. So you can choose which is right for you, and not miss anything important. The Item restriction is in effect for each one though. So keep that in mind when using magic. That being said, all of these enemies have been battled through the game up to this point. And they are the same strategies used to defeat them.

Level 1 Cloud|Aerith|Tifa|Barret Vs. Wild Animals

This round is not terribly difficult. None of the enemies have any major attacks to watch out for either. The best bet is to get a Restore Materia or Chakra Materia to heal if needed. Then get a Lightning + Magnify Materia. Lightning does not miss, and can easily wipe out the field, if not remove weaker enemies. Makes the round a lot easier. You can also equip a Wind Materia for the final confrontation with the Lesser Drake.

Round 1: Wererat x2

Round 2: Blugu x1, Gorger x2

Round 3: Guard Dog x1

Round 4: Hedgehog Pie x1, Grashtrike x1

Round 5: Lesser Drake x1

Battle Challenge	Reward
Cloud vs Wild Animals	Manuscript: Legacy: Ascension
Barret vs Wild Animals	Manuscript: Legacy: Catastrophe
Tifa vs Wild Animals	Manuscript: Legacy: Dolphin Flurry
Aerith vs Wild Animals	Manuscript: Legacy: Planet's Protection

Level 2 Two Party Members Vs. Slum Outlaws

When humans are the prey, fire comes out to play. Equip one of your two members with a Fire Materia, to remove weak enemies really fast. Be careful of the standard Bandit's electric fields. It will stun whoever steps into it. If this occurs, switch to your secondary player, and continue on.
When confronting the Huntsman in the 4th and 5th rounds, be sure to get behind him for heavier damage....or nail him with Fira(ga).
It should be noted that there are two different kind of bandits in these rounds. (Also appear in the Underground Expressway in Chapter 13 & 14) Distenguished by their longer white and red face masks, they have less HP than the standard Bandits, but the stats are the same. They will not steal from you though, that I have seen.

Round 1: Corneo Lackey x2, Bandit x1

Round 2: Corneo Lackey x1

Round 3: Corneo Lackey x1, Bandit x2

Round 4: The Huntsman x1

Round 5: The Huntsman x1, Bandit x4

Battle Challenge	Reward
2 Party Members vs Slum Outlaws	Clarity Pendant

Level 3 Cloud|Aerith|Tifa|Barret Vs. Shinra Thugs

In this set of fights, it is a great idea to bring in both fire for the humans, and lightning for the machines. It is difficult to say which should also have a Magnify set with it. My suggestion is have Lightning + Magnify, while having Fire + Elemental on your armor. If you have Fira or Firaga, the Flametrooper cannot do any major damage to you with his flamethrowing attacks. Bring in Chakra to keep healed if needed.
Remember the Riot Trooper is best attacked physically from behind, or will fall fast with a Fire spell or two. The only real challange is the Sweeper, but a few higher tiered Thunder spells will get him down fast.

Round 1: Security Officer x1, Grenadier x2, Sentry Gun Prototype x1

Round 2: Flametrooper x1

Round 3: Shocktrooper x1, Sentry Ray x2

Round 4: Riot Trooper x1, Slug Ray x2

Round 5: Sweeper x1, Sentry Launcher x1

Battle Challenge	Reward
Cloud vs Shinra Thugs	The Art of Swordplay Vol. II
Barret vs Shinra Thugs	Sharpshooter’s Companion Vol. II
Tifa vs Shinra Thugs	Way of the Fist Vol. II
Aerith vs Shinra Thugs	Telluric Scriptures Vol. II
3 Member Team vs Shinra Warriors	Tarot Cards

Level 4 Two Member Team Vs. Team Payback

For this final round of battles, be sure to definitely bring in Ice. A good chunk of the enemies are weak to ice magic. Setting it to someone's weapon with an Elemental Materia will do greatly as well. Bring in Breach, to aid in the first round with the Hedgehog Pie King. Lightning will aid with bringing down the Chromogger and Jury-Rigged Cutter. When it becomes staggered, you can break off its arms to decrease its offensive attacks. Remember to keep your distance with the Bombs, when they are doing flame attacks, and Mark II Monodrives need to be hit with magic in order to break their physical barriers.

Round 1: Hedgehog Pie x1, Hedgehog Pie King x1

Round 2: Wrath Hound x1

Round 3: Chromogger x1, Doomrat x1

Round 4: Cerulean Drake x1, Bomb x1

Round 5: Jury-Rigged Cutter x1, Mark II Monodrive x1

Battle Challenge	Reward
3 Member Team vs Team Payback	Moogle's Amulet