Well, it wasn't just the eye color, it was the bands in the tail, the distance the feathers do or do not travel down the legs, the color of the legs, the breast color & amount of streaking, and the SIZE. Sharp shinned are small hawks whereas Cooper's hawks are medium sized. Often we have nothing in a photo to give us size perception but in this case, the chain link fence was great for getting an idea of the size. Also, it's head was turned which allowed a look at the classic coloring of the head/neck of the sharp shinned & the sharp shinned are here this time of year.
I once read something about IDing birds which is the most helpful thing I've ever learned in order to ID them. When you see them, do not think about what they are; just LOOK & take a mental photo, notice leg color, beak color, breast, back, wings. SIZE, BILL SHAPE, tail -- is it banded, rounded or forked. Just look beginning at the head & work your way down trying to mentally remember as many features as you can. That is the single most invaluable thing I've learned. And it seems like soooooo DUH!!! But before I read that, I was not looking that way. I would try to remember every feature of the head & zip! the bird would be gone before I got any further. Just act as if your eyes are a camera taking photos in rapid succession, click, click, click. click. The more you get, the better chance of IDing it.
