I think you are right. It looks like a young Cooper's Hawk to me based on the yellow eyes, size, and thick vertical chest streaks. Sharp-shinned (Accipiter striatus) and Cooper’s (Accipiter cooperii) hawks have a lot of variation in plumage and size and there no single field marks that distinguishes one species from the other.
Sharp-shinned Hawks average 10-14" long. Females are larger and can appear nearly as large as a male Cooper's Hawk. Where Cooper's Hawks average 14-20" long and males are smaller and can appear nearly as small as a female Sharp-shinned Hawk.
The following are ID tips from Project Feeder Watch for the Sharp-shinned Hawk and Cooper's Hawk:
1) The tail feathers of Sharp-shinned Hawks appear squared, whereas a Cooper's Hawks tail looks rounded.
2) Cooper's Hawks have a barrel shaped chest. Sharp-shinned Hawks are widest at the shoulder and get distinctly narrower down to the hips.
3) A Sharp-shinned Hawk's head looks small compared to the body, and a Cooper's Hawk's head looks large.
4) Cooper's Hawks are usually larger than Sharp-shinned Hawks.
5) A juvenile Sharp-shinned Hawk has very thick, rufous stripes that extend down the lower belly while the Cooper's Hawk has very thin, dark vertical streaks that fade away on the lower belly.
6) Sharp-shinned have very thin toes and legs, compared to the Cooper's Hawk.
7) The color of the nape of an adult Cooper's Hawks is pale with a clear contrast to a dark cap. Juveniles of both species can show a pale nape, however.
More identification tips and challenges can be seen on Project Feeder Watch's Accipiter Photo Gallery page: http://www.birds.cornell.edu/pfw/AboutBirdsandFeeding/accipiterphoto.htm
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