Unfortunately Ancestry's way of presenting military records obliges the average person to make a leap of faith (or leap of reason).

Let me explain why I asked for a name and residence. Almost all men who enlisted during the war joined a company being recruited near their home. For that reason, knowing a soldier's name and county will usually help identify his command. It looks as if War Department clerks weren't sure what to do with records of this command and decided to create a separate file for Capt. Cook's Horse Artillery. Records of about seventy members are found on reel 94 of National Archives Microcopy M269, Compiled Service Records of Confederate Soldiers Who Served in Organizations From the State of Mississippi.

Since Greenville Cook was 1st Lieutenant of Bradford's Battery (Co. "F", 1st Mississippi Artillery), I am confident we can associate the two commands. Bradford's Battery, which evidently became a horse artillery company after the fall of Port Hudson, is mentioned several times here --
http://mississippiscv.org/MS_Units/1st_mississippi_light_artillery.htm

Copies of these service files can be ordered here --