When registering for the Public Library, an email is sent to your account to allow you to set a password.
Gmail has a tendency to put this email in a Promotions folder (or sometimes spam).
If you don’t see the email in your inbox, please look check those locations. If you find the email in spam, mark it as Not Spam. If it’s in the Promotions folder, drag it to the inbox.
This is where the Promotions folder is located: