Sounds like both see the other as the parasite that is burning out existence.while hydelyn didn't use the exact words, it seems a similar reasoning for the banishment of Zodiark.
One fear is that the continuing cycle will cause Zodiark's revival and an end to existence as we know it. Could the other be that stopping the cyclical nature of destiny is what causes the end of existence? That without Zodiark's supposed revival there is no one to take the seat of "the mothercrystal" after Hydelyn fades totally?
/speculation
