Hayabusa is Shan-Yu's pet falcon. He is Shan-Yu's eyes and ears over China. When someone sees Hayabusa, it probably means that Shan-Yu and his army are coming. He originates from Disney's Mulan.
During the boss fight with Shan-Yu, Hayabusa appears along Shan-Yu. He can attack you while you are attacking Shan-Yu, so you should kill Hayabusa first.