emails in Time Machine with seagate backup plus

In general, most users are accessing emails using IMAP protocol, so they can be retrieved and read from multiple devices. if that is the case, the INBOX is NOT stored "on My Mac".

Using IMAP, the INBOX is on the Service Provider's mail server on the Internet, and messages are retrieved from there directly. Once they are gone from the service providers INBOX and Trash, they are GONE forever.

In general, only files stored in user-created mailboxes "on My Mac" are included in your Time machine and other backups.