Earth's rotation
As a payload is lifted up a space elevator, it would gain not only altitude, but horizontal speed (angular momentum) as well. The angular momentum is taken from the Earth's rotation. As the climber ascends, it is initially moving slower than each successive part of cable it is moving on to. This is the Coriolis force: the climber "drags" (westward) on the cable, as it climbs, and slightly decreases the Earth's rotation speed. The opposite process would occur for descending payloads: the cable is tilted eastward, thus slightly increasing Earth's rotation speed.