Bullseye is the name of Woody's horse. Woody and Bullseye are
reunited after Woody was kidnapped and taken to a museum (i think
it's a museum at least) where he meets the other three members of
his original t.v show ; Jessie, his sidekick, Bullseye, his
faithful steed and Stinky Pete, who I have no idea who he is, but
he isn't very nice :]
Hope it helped
Jess XxX :]