First, I am a mathematics major who is doing an REU (Research Experiences for Undergraduates) in mathematics at my school. Initially, I wanted to do an REU outside of my school because I have had this professor before, and being in his class is difficult, but not unbearable. Right now, it is unbearable. For the first project, he has asked me to prove a proposition in category theory, and his criticism is not helpful nor is it actually advice. He has went as far as to say "this is your project, not mine", even though my proof is actually correct for the problem at hand; I have seen other verified proofs and had mine verified by mathematicians. It just makes me feel stupid, and I feel depressed and stressed coming to school everyday. What are my options in this situation and what is the most ethical?
Your professor is not being "unethical". It's possible he's a difficult person but ethics is not a part of this.
There's a pretty good chance that if your professor says your proof is not correct, that it really isn't. And he may not be as unhelpful as you think. Students sometimes think an instructor is unhelpful if they don't give them the answer any time they're stuck. But simply giving you the answer won't make you a better mathematician. Forcing you to keep plugging at it until you find the answer on your own will make you a better mathematician.
My advice: Consider the possibility that he's right, you're not and that he wants you to keep working at it until you solve it on your own and go from there.
It’s really hard to understand your question. In your questions and comments you’re using language in imprecise and confusing ways. I’d imagine that when you try to explain math you’re probably also explaining it in imprecise and confusing ways. My best guess is your professor genuinely doesn’t understand what your argument is. Have you tried writing it down carefully line-by-line and thinking through whether each step is totally clear and explained precisely? It also might help to try to explain it to another student.