Does your town or county have the feline equivalent of "leash laws" which require cats to be on their owners' property or under their control? I doubt it, but IF there were such a law, it could be used to rationalize the trapping of cats.
Did your cat have a collar or ID chip? If so, that would be proof of being owned, as opposed to being feral (not to mention basic reaction to humans).
Was the trap baited? If so, the cop INDUCED the cats into his yard, as opposed to his trapping nuisance animals.
Putting all the above aside for the moment, even IF all the above questions were answered in the cop's favor, that does not excuse his taking the captured animals to HIS house in another city. At best, he would be authorized to bring them to the AC officer, have the animals logged in and placed in a shelter.
What he did appears to, at minimum, trespass to chattels; presuming the trap was baited, combined with taking the cat(s) to his house in another jurisdiction suggests theft.
Your difficulty is proving he took your cat, that he knew/should have known the cat was owned, and your damages. Without being able to prove all three, your case is weak, if not untenable.
If you can prove the first two, you still have the basis for lodging a complaint against the officer. I suggest contacting the Ohio and local bar associations for referrals to attorneys in your area, once you have compiled the necessary documentation.
The foregoing is for general information purposes and does not establish an attorney-client relationship.
Your scenario doesn't make sense - what do you mean he trapped the cat in his yard but then took him to his house in another city? How do you know this happened? Was he acting as an animal control officer or as a private citizen?
Aside from all of that, I agree with my colleague. I would also ask: how would he know the cat was your cat? Why was your cat wandering loose without a collar, tags, or microchip? Have you had complaints about your cat being loose? Most importantly, what are your damages? At most you would be reimbursed for the replacement value of your cat (and that's "retail" value, not what you think he is worth). You should speak with a local attorney for further guidance.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.