Since he stayed in Hosokawa no sho estate in Harima Province (present Miki City, Hyogo Prefecture) and become a feudal lord, he was conveniently called Tamezumi HOSOKAWA, but he had no relationship with the Hosokawa clan, the shogunal deputy of the Muromachi bakufu (Japanese feudal government headed by a shogun).
長らく播磨国細川庄（現在の兵庫県三木市）に下向して領主化していた事から便宜上細川為純（ほそかわためずみ）といわれることもあるが、室町幕府管領の細川氏との関係はない。