Where is the sealed cave in Final Fantasy 4

Head back to the Dwarven Castle. In the foyer, head up the stairs on the left, then go down the stairs in the top right corner, & then go through the first door you see to get to the infirmary. Once there, talk to Cid, who is revealed to have survived. He will fix your ship so that you can cross the magma. Fly southwest of Dwarven Castle, and there will be a cave surrounded by land. That is the Sealed Cave.