how do cold and warm fronts work
If the cyclone picks up a cold air mass along its polarward edge and drags it along, around its Eastern or Western edge, we have a cold front moving equatorwards. Vice versa, if it picks up a warm air mass along its equatorward edge and drags that along, we have a warm front moving polarwards. Notice that since every front involves cold and warm air masses, the distinction between a cold front and a warm front is which one is moving and pushing the other one away.