The latissimus dorsi is actually the primary mover when performing pull-ups. Most believe the biceps are the secondary movers but they are actually 3rd in line. The pectorals are the secondary ones.
Starting from a full hang, the latissimus dorsi initiates movement from zero to approximately 45 degrees. The pectorals assist during the next 90 degrees and the biceps become fully active during the final 45 degrees.
The point is if your goal is to increase your productivity in performing pull-ups, then you should start doing pull-ups supersets with the bar dips and pull-overs to improve performance.