I hear something halfway between kozee and kozay/kozé. Is that right?
Your ear does not lie. As a french I hear it between é and i also. And it is : https://cs.wiktionary.org/wiki/kozy
The czech y is pronounced [I] https://en.wikipedia.org/wiki/Near-close_near-front_unrounded_vowel
In the IPA vowel diagram you can see it is right in the middle also. https://en.wikipedia.org/wiki/Vowel_diagram
But when it is the long form it goes back to our classic i : https://cs.wiktionary.org/wiki/vysok%C3%BD
I hope I am not saying wrong informations as I am not a czech specialist.