I have just noticed an issue with Exchange 2010 based hybrid and AADSync. On-prem AD account has assigned on-prem mailbox as well as remote mailbox (Exchange Online – O365). Yes, it seems to be possible, but only technically, because mailbox functionality is broken from end-user point of view at this point.
According my investigation it was caused by wrong procedure by mailbox creation. SD created remote mailbox , ran AADSync, disabled remote mailbox, forgot to run AADSync and created on-prem mailbox immediately.
So what to do then? We have to ensure only one mailbox either in Exchange Online or Exchange On-prem so based on our needs.
I decided to keep on-prem mailbox, because it was used by affected end-user.
Well, I exported all mailbox data from remote mailbox to PST file. There are a few possibilities how to do it e.g. via Outlook or eDiscovery search. I wanted to expert all available mailbox data including dumpster so I preferred eDiscovery approach (be aware to have mailbox still enabled during that time).