Woodward's 15th KY Cavalry (its proper designation from the CS War Department) did fight with Forrest on the West Tennessee raid of December 1862.
Woodward raised the Oak Grove Rangers in May 1861 which became at least two companies of Helm's 1st KY Cavalry. These were one year units both. When the enlistment expired, Woodward raised Woodward's KY Cavalry Battalion and with that helped capture Clarksville, TN but was repulsed at Dover in August 1862. This unit fell apart and in December he raised the 2nd KY Cavalry. When the paperwork got to Richmond his unit was renumbered as the 15th KY Cavalry as John Hunt Morgan had already raised a 2nd KY Cavalry - hence the correct designation being Woodward's 15th KY Cavalry.