If a mailbox is offline in Mail on Mac
When an email account is offline, a lightning bolt icon appears next to the account’s mailboxes in the Mail sidebar and at the end of the Favorites bar.
In the Mail app on your Mac, do one of the following:
Take all accounts online: Click the lightning bolt icon or choose Mailbox > Take All Accounts Online.
Take a specific account online (if you have more than one account): Click the lightning bolt icon next to the account’s mailbox, or choose Mailbox > Online Status > Take [account] Online.
If you can’t take an account online, make sure you’re connected to the internet. If a warning icon appears next to the account name, click it to open Connection Doctor. Learn more about using Connection Doctor.
Some mailbox names can appear dimmed if you don’t have permission to access them, such as public mailboxes on a Mail server.