Yes, it will just happen. It will grow if you take more to add to it but what is there will turn in to lovely black compost
In summer it happens so quickly we often get 3 batches and that's just because we're too lazy to empty the bin out more often. In autumn and spring its slower and in North American winter I am guessing it doesn't happen at all.
Worms help but they will arrive on their own. We find that larger larvae like Pachnoda sinuata
will chomp through huge amounts of raw material to produce stunning black compost in amazingly short periods of time.