Were the Huns and Mongols at one point rivals?

Who was Genghis Khan’s main rival?

Jamukha

Jamukha (Mongolian: Жамуха; Chinese: 札木合) was a Mongol military and political leader and the chief rival to Temüjin (later Genghis Khan) in the unification of the Mongol tribes.

How many babies did Genghis Khan have?

four sons

In this context it’s pretty obvious, the Mongol Empire was the personal property of the “Golden Family,” the family of Genghis Khan. More precisely this came to consist of the descendants of Genghis Khan’s four sons by his first and primary wife, Jochi, Chagatai, Ogedei, and Tolui.