IT should but that depends on how and where it broke.  The easiest test is
to check the output temp first thing in the morning with the engine dirt
cold before the sun or the heater core has a chance to warm things up.  Set
it to full cold and start it up and check the air temp.  If the airflow is
cooler than the outside air, then you probably have a problem with the blend
air door.  It may be broken at the cable attachment point where the cable
can push it to the hot side but cannot pull it past the midpoint to the cold
side.  It could also be some crap plugging up the evaporator fins and not
letting the air pass through it but if that is the case, I would suspect
that you would see a reduction of airflow at the full cold position.