Do hawks eat coyotes

Yes, coyotes do eat hawks. They also eat rattlesnakes, mice, and rat. The only way to eat birds such as hawks is if the animal is injured and resting on the ground or somewhere where its in the predators reach.
Under normal circumstances a coyote could not catch a hawk. However, an injured hawk, an immature hawk, or a dead hawk might be eaten by a coyote. They are known to target any bird that nests on the ground, so nesting hawks are generally safe.