Final answer:
The nerve in question is the thoracodorsal nerve, accompanying the thoracodorsal artery, a branch of the axillary artery that supplies blood to the latissimus dorsi muscle.
Step-by-step explanation:
The nerve that is found on the deep side of the latissimus dorsi at the humeral end, which serves that muscle and runs with an artery of the same name, is the thoracodorsal nerve. This nerve is associated with the thoracodorsal artery, which is a branch of the subscapular artery, itself a branch of the axillary artery. The axillary artery is a continuation of the subclavian artery as it penetrates the body wall and enters the axillary region. It supplies blood to the region near the head of the humerus via the humeral circumflex arteries. The majority of this vessel continues into the brachium as the brachial artery.