I think others told the important part. Just know he can take PFL within one year of the birth so if he wants, he can stay on unemployment til it runs out and take PFL then.
You just apply for PFL and stop certifying benefits under unemployment
You have to close out your claim for unemployment and then apply for PFL. As far as the small details I’m not sure because I’ve never done it but you can try calling or you can go through your state assembly person and a representative will open a case for you and contact EDD on your behalf. They usually get everything done same day. https://findyourrep.legislature.ca.gov/