Greenville Cook was a Sr. First Lieutenant in the First Regiment Light Artillery in Company F. He was the only "Cook" in Company F. He then became Captain Cook of Cook's Company Mississippi Horse Artillery also a part of Company F.
Although the 1st Regiment was designated as Light Artillery, the historical records indicate that they function as heavy artillery also.
I do not see a reference to Bradford's Battery in his records, but one may be a subset of the other. I am not sure in the Conferderate Army what the different designations are. Perhaps a military historian can give us some help in decifering the Regiment: First, Company: F, Battery? Company?