Further research has yielded an interesting result.

The First Regiment of Mississippi Light Artillery was also known as Wither's Artillery. If you put his name as Wither's Artillery into Google you will get many interesting results about the military pursuits of his soldiers.

Company F was also known as Bradford's Battery or Bradford's Company.

The terms as written seem to be somewhat interchangeable although I suspect that one is correct and Captain Bradford if he were still alive would quickly correct the Company's name. Captain Cook was a part of his Company. They make have designated Captain Cooks company off as horse artillery.