W. A> Broadway was a member of Co. "F", 37th Mississippi Regiment. All Confederate officers and men surrendered to Federal authorities at Vicksburg were paroled and allowed to go home. General Pemberton also got an agreement that none of his men would be allowed to take the oath of allegiance. Of course Broadway could have done so later, but the oath had nothing to do with his ability to go home after Vicksburg.
If this man took the oath of allegiance at some point, the paper should be in his military service file.