The answer is that Mihawk is a Nerona, he was trained on the sword by Garling and Nasujuro when he was young but decided to roam the seas in search of strong swordsmen, he was groomed to succeed Garling as the commander of the God Knights.
He will return to his family EOS, but they will reject him as they are in a possession of an upgraded version of him, the seraphim S-Hawk. So Mihawk has to defeat him if he want to come back so he fights S-Hawk and is overwhelmed but gets a haki bloom and defeats S-Hawks killing him and ascending beyond Ryuma’s level as Zoro’s EOS fight