Believe there was a misprint on his record, and it was the 1st Kentucky Battery to which he
was assigned.

Two months later, he deserted the Federal army. This was quite common for Confederate soldiers who joined the Yankee army, "Galvanized Yankees", and subsequently soon thereafter deserted them.

John F. Williams

Residence was not listed;
Enlisted on 9/8/1863 at Louisville, KY as a Private.
On 9/10/1863 he mustered into "Batty C" Co. KY 1st Light Artillery
He deserted on 11/16/1863 at Louisville, KY

Sources used by Historical Data Systems, Inc.:
- Report of the Adjutant General of the State of Kentucky

See also: https://en.wikipedia.org/wiki/Battery_%22C%22_Kentucky_Light_Artillery