The rear derailleur is the mechanism that shifts the chain at the rear sprockets, and it uses a cage with two pulleys: the lower tension pulley and the upper guide pulley. As the rider turns the shifter, cable tension moves the derailleur linkage sideways, and the chain is guided onto a new sprocket while the return spring moves it back when the cable is released. Smooth shifting comes from timing and light load: shift while pedaling, do it early before a hill, and ease the pressure if you must change under strain.
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