If you were asked what the function of the latissimus dorsi was, what would you say? Would it be adducting the arm and extending the arm - as occurs during a pull up or DB row? Would you take it a step further an add that they act as a humeral medial rotator? You could even add the eccentric function in which they decelerate the arm into flexion and lateral rotation. That would be a start.
The lats don't just directly affect movement of the arm. Their function and influence on posture is more complex. The proximal attachment of the lats is onto the last 6 thoracic vertebrae, inferior 3-4 ribs, external lip of the iliac crest, the sacral vertebrae via the TLF and the inferior angle of the scapula (but I think that's been debated).
In blending with the TLF, the lats contribute to the stability of the lumbo-pelvic-hip complex, core, whatever you like to call it. If you think of the LPHC as one big joint (in which we need to remain stable in order to transmit force through it), the lat (right or left) and glute-complex (right or left) lay contralateral to each other and assist in stabilization of the LPHC by creating tension through the TLF. This myofascial linkage is reason the mechanics of the shoulder girdle rely so much on the "health" of the contralateral hip and foot/ankle complex.
Other muscles blend into the TLF as well including the internal oblique and tranversus abdominus. If nothing else this demonstrates how much of a role the lats have on core stability. If one member of this "team" isn't fulfilling its role, the "team" suffers and core stability is compromised. As picky as this all may seem, most good squatters consciously pull the bar into their traps by contracting their lats as hard as they can. Good benchers understand the importance of the lats in creating a stable base to push from, whether they care about myofascial linkage or not!