This is inspired by this question about the equivalence between the category of finite sets and non-negative integers. Now this question was (rightly, I guess) closed, but the fact was surprising to the OPer. I didn't know about this, but it was easy to verify. I had a little bit of difficulty understanding the nuances between categorical equivalences and category isomorphism until I thought about the analogy with homotopic spaces and homeomorphic spaces.
I was wondering about other equivalences that might unexpected and/or not so straightforward to prove. I would appreciate answers to a general audience...things that a beginner or an expert might find of interest.