Adopted. Cao Cao is a descendant of Xiahou's family? So you see that Cao Cao and Xia are very close.
Cao Shi and Xiahou's family are both big families in Peiguo Qiaoxian County (Bozhou City, Anhui Province). However, since Cao Teng, Cao Shi's family has prospered, and has become a big official in the imperial court and counties. Because Cao Teng was deeply trusted by the general and Emperor Huan of Han Dynasty, his adopted son Cao Song was Qiu. Brother Cao is the satrap of Yingchuan, son Cao is the assistant of Chiguan and a captain of Changshui; Another cousin, Cao Ding, is an official and an official.
On the other hand, although the Xiahou family has a great career, after all, no one in their ancestors has ever been an official. He is only an indigenous landlord, not an official.
In order to gain the development of power, Xiahou gradually moved closer to Cao Shi. Although Cao Shi made his fortune as a eunuch by Cao Teng, he had a good reputation among literati and was neither arrogant nor impetuous in his hometown. So, Xia Houdun began to move closer to Cao Shi, and Xia Houdun and Xia, two brothers (Xia is Xia Houdun's younger brother, but at least not the same grandfather), took a fancy to Cao Cao, who had a good reputation in the mouth of the literati.
This Xiahou family is a descendant of Xia Houying, a hero of the Western Han Dynasty.