Before you travel back to the Forgotten City you need to make sure that you have picked up the Key to the Ancients. The Key can be found in the ocean and is obtainable after you have completed the Underwater Submarine Battle and received the Shinra Sub as a vehicle.
It is located in the northern section of the ocean underwater in a small cave. Use the map and the screenshots below to help you find it:
Jump back into the Highwind and fly it over to the Forgotten Capital / Forgotten City. You do not need to travel through Bone Village or the Sleeping Forest this time as you can land the Highwind right next to the Forgotten Capital on the world map (as shown in the screenshot below).
Take a left at the fork in the road and follow the pathway around until you reach the Forgotten City.
Bugenhagen will immediately begin talking once you reach the open area at the end of the pathway.
Travel around the pathway to the right side of the screen and proceed off screen to automatically reappear at the top of the stairs. Walk down and speak to Bugenhagen to continue the story.
Exit the fountain and then exit the Forgotten City. There will be a short cutscene involving Rufus, Scarlet and Heidegger back in Midgar with the Sister Ray as you exit.
Proceed back to the world map to continue the game. Another short cutscene will take place when you try to board the Highwind.
Your next battle is against Diamond Weapon but you cannot actually fight Diamond Weapon while it is out in the middle of the ocean. Bumping into it with the Highwind will just cause Cid to scream at you for attempting to break the Highwind and will suggest that you attempt fighting it on the ground.
In preparation for the battle make sure that you equip a character with 
Steal Materia and equip accessories that either prevent status effects, such as your 
Ribbons, or that nullify fire-based magic, such as your Fire Ring.
Equip powerful Magic Materia such as 
Ultima Materia, Contain Materia and Comet Materia to each of your characters so that they all have access to non-physical attacks.
HP: 30,000
Reward: None
Diamond Weapon starts off the battle with immunity to physical attacks. Focus on using high powered spells such as Ultima, Contain and Comet. Diamond Weapon also has a weakness to lightning-based magic making Bolt3 from your Lightning Materia extremely effective.
Make sure that you steal a new weapon for Yuffie called 
Rising Sun at some point during the battle.
The most powerful attack you will face is called Diamond Flash and there will be a countdown from 3 before Diamond Weapon will use it. The attack will reduce all of your character’s HP down by 7/8ths but cannot actually kill you.
A video demonstration of this battle is included below for reference:
Diamond Disintegrator
You will receive the Diamond Disintegrator Trophy once the battle against Diamond Weapon concludes. You won’t actually defeat Diamond Weapon but you will receive the trophy at the end of the battle.
Diamond Weapon will run away before you actually defeat it. There are quite a few cutscenes and FMVs afterwards.
Travel back to the Northern Crater once you are back in control of Cloud in the Highwind. The energy barrier that Sephiroth had created has now been completely removed.
The next mission is to stop Hojo by traveling back to Midgar.
Note that this is the last point in the game where you can still complete the Wutai side quest and obtain the 
MP Absorb Materia. Do that side quest now if you have not done so already!
Proceed to the next page to return to Midgar for the final time…
