The pulling hand when punching, blocking and striking is very important. As important you may say as the hand doing the technique itself. The pulling hand or hikite as it's called in the dojo contributes much to the effectiveness of a technique, and the more advanced in karate you become, the more you will appreciate its significance. When, for example punching, not only should you place emphasis on throwing that arm out, but you should also concentrate on returning the other arm as fast and as powerfully as possible. This is vital in making your karate work to its maximum potential. If your right arm is punching, then your left arm is preparing (or has done the previous punch), then this is the fist being returned to your hip.
Different associations place a different degree of attention on the returning arm Some associations feel that you should seriously think about retracting the arm back to the hip as fast as possible in order to help generate power in the punch being fired out. Others feel hikite should simply be a natural reaction to the punch going out

Stage 1: Your arm is out as a preparation or as a punch.

Stage 2: Your returning arm begins to return, pulling back to your hip..

Stage 3: Your returning arm continues to pull back to the hip.

Stage 4: The arm completes its destination at the hip. This is hikite.