According to the OH and KY reciprocity only your resident state would tax your income earned in the other state. There will not be any credit for taxes paid to another state for double taxation. When you work your part-year returns for each state you will allocate your wages according to your resident time in each state. Therefore, if there were too much tax withheld by KY based on your allocated residence time wage amount you will get a refund from KY (not a credit from OH).
Likewise for your OH state interview. Your OH tax will be based on the wage allocation you make in the state interview. You do not claim a credit for taxes paid to another state.
Your employer likely aided your allocation computation by splitting your W2 box 1 amount between the two states. On your W2 the line with KY in box 15 should have a box 16 amount that is less than the box 1 amount. The OH box 16 amount should make up the difference. If this is the case, the allocation will be done for you by your federal "Wage & Income" W2 entries and you will simply verify in the state interviews.
If for some reason the W2 was not allocated properly you will make the allocation adjustments in each state interview.
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