Ok so I have a theory. What if Lillie is AZ's daughter? This makes sense BECAUUUSE AZ lost his floette that he loved dearly. And it was a flower. Lillie is the name of a flower. What if AZ named Lillie after a flower to remember his floette? ALSO AZ was devasted by the pokemon war that resulted in the death of his floette. And as we know, Lillie is against pokemon battling, probably because her father told her the story of how pokemon can die from it. Also maybe AZ is a traveller because he travels from Kalos to Alola to visit his daughter. Also AZ resides in Kalos for the most part, which is where Zygarde resides as well. And since it's basically almost confirmed that Lillie is related to Zygarde, what if it's because her father made that connection? But that's just a theory.