The answer above is still correct. Essentially the reciprocal state rules don't apply when you move from one state to the other. You have to file a part year resident return for both states. Your employer did the withholding correctly so that will make it a little easier, but part year filing is always messy. Do the OH return first.
For 2022 you will be a full year KY resident and reciprocity applies (you don't file OH for 2022).
The reciprocity agreement does not apply to local city taxes. You are not required to file an OH city return (starting in 2022), but you are also not entitled to a refund of the city withholding (box19 on your W-2), unless there was an error in the amount. If you are subject to a KY county or city tax, you should check to see if the KY city will give you a credit for the OH city tax withheld. KY does allow an itemized deduction (Schedule A), on the state return, for local tax paid.
