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• For incoming mail, on your server run a mail retrieval agent like fetchmail to fetch mail from the externally hosted mailbox into a maildir on your server.
• To serve that maildir to your clients, on your server run a mail delivery agent like the IMAP server Dovecot.
• To accept outgoing mail from your clients, on your server run something like Postfix with a relayhost configured with the details of your externally hosted SMTP server.

There’s nothing unusual or tricky about any of this arrangement.