Alright, before we go to Sylph Cave again, let head our way to Fabul castle because we need something for Yang. Remember he is still sleeping in Sylph's house?

Just follow the map, once you are in the castle, go to the left tower where Yang's wife lives. She'll then give you something that can wake up Yang.

Continue our journey to Sylph Cave in the hollow Earth. When you reach the Sylph house, give Yang a good wake up frying pan... lol.

Now Rydia learned how to summon Sylph and Yang gave you Brace, Kick, and Focus augment.

Augments:
Give Kick and Brace to Cecil, again this is not required but recommended.

And before you exit the cave, you should notice another Eidolon in room #6. So go ahead and talk to him.

Since you borrowed the frying pan from Yang's wife, you should return it back, right? Now going back to Fabul castle and visit Yang's wife. Return the frying pan back and she'll appreciate it and gift you a knife. Yeah! seriously a knife? well, this knife is kinda useless as a wielding weapon, but with Edge who has the ability to throw a weapon, he can make use of it and deal 9,999 damage :D.