A common issue seen in powershell when running the cmdlet get-mailbox -arbitration
Warning: The object account
Warning: Database is mandatory on UserMailbox
In English: this means that the aribitration account in question does not have an associated mailbox. This can occur for many reasons but the more common is because at some stage the last Exchange 2010 server had been removed forcibly from the organisation.
The fix firstly is to disable the arbitration account in question. Do this by typing the below command subtituting "SystemMailbox{1Fe78u....}" with the arbitation account you need to fix.
Disable-mailbox "SystemMailbox{1Fe78u....}" -Arbitration -DisableLastArbitrationMailboxAllowed
Next re-enable the mailbox for the arbitration account which will in turn create a HomeMDB reference for the account. This command will create a mailbox on a random mailbox database. You can target it using the -targetdatabase reference in the command if you so wish.
Do this by typing:
Enable-mailbox "SystemMailbox{1Fe78u....}" -Arbitration
To verify your arbitration account is now fixed type the following:
Get-mailbox -arbitation. This should return an entry for the your previously problematic arbitration account.