Integrate mail system between O365 and mail system on-premise

First thing you need to do is to find out what On-Premises e-mail system you customer has...if customer has Exchange 2010 / 2013 / 2016 / 2019 then this scenario is fully supported under a hybrid deployment so you can do what you have been asked for. But if your customer does not have Exchange, the hybrid request cannot be done in the same way and you will need to evaluate what your options are here such use e-mail redirection for migrated users, partially domain validation in Office 365, etc