If you are reading this you may have seen my previous blog on fixing the Exchange Online error for not being able to send emails to a user from an external address. To link the two objects you will need to run the following commands:
This first command is to view the entire forest and then enable a remote mailbox for the user, if they already have one you can skip to the next sets.
Set-ADServerSettings -ViewEntireForest $True
Enable-RemoteMailbox -Identity email@example.com
This next command you will need to run in Exchange Online PowerShell:
Get-Mailbox -Identity firstname.lastname@example.org | FL *GUID*
Finally you will need to go back to the Exchange on-premise PowerShell and run the following:
Set-RemoteMailbox -Identity email@example.com -ExchangeGuid <ExchangeGuid from Exchange Online PowerShell>
Hope this helps!