I got new car insurance, but my old insurer is still expecting me to pay, can I just ignore them or do I have to do something?

I was using Direct Auto and bought a new car and my insurance went way up, so I went shopping around and ended up finding a much better price at Liberty Mutual for the same coverage. I switched the Liberty Mutual and even paid for my first term a few weeks ago, but I set my coverage to start the day before my next payment with Direct Auto, which is today, so my Liberty Mutual coverage began yesterday.

Liberty Mutual said it will contact your former insurer for you to ask for cancelation, but today I got an email from Direct Auto saying my insurance payment is due. Is there anything I have to do? Or is Liberty Mutual handling it and I can just ignore it?