Why does adverse yaw exist?
Indeed, when an aileron moves upward, it locally generates less lift and less drag.

Assume we are talking about the aileron on the right wing. 
The reduced lift drops that wing, rolling the aircraft right. But the reduced drag accelerates that wing, yawing the aircraft left. 

On the left side the opposite happens, but it has the same effect. The aileron moves down, increasing lift to roll the aircraft right. But the added lift increases drag, decelerating the left wing to yaw the aircraft left. 

That is the adverse yaw effect.