Why the ulnar nerve innervates the finger adductors?

The fingers are the post-axial fin ray bones, and the ulnar nerve is the post-axial nerve. During evolution, the post-axial fin rays have bent forwards, since the central axis of the fin has bent at the hand. The rest of the fin rays have disappeared. So this is why the post-axial nerve (the "ulnar" nerve) innervates the finger adductors. Since the pre-axial fin rays are all gone (except maybe the thumb, it is sometimes ascribed to pre-axial bones), the limbs are said to have undergone "posteriorization". This is also why the pre-axial nerve, the "musculocutaneous" nerve, does not innervate muscles in the forearm or hand - our arms are mostly the post-axial (caudal) part of the fin of the fish, and exactly why that is, adaptations for crawling locomotion explains that.