In a drum hoisting system through a vertical shaft, over-winding—where the cage or skip is raised beyond its intended stop point—is prevented by a detaching hook. This safety device is designed to disconnect the cage from the hoisting rope if it travels too far, preventing damage or accidents. The Lilly controller regulates speed, the caliper brake slows or stops the drum, and the safety catch secures the cage in case of rope failure, but none of these directly address over-winding like the detaching hook does.
Thus, the correct answer is Detaching hook.