The Canvas Inbox is an internal system for messaging in Canvas, that students and teachers may use to communicate. It is important to understand that it is not the same as email.

Notice of new message per email

When someone sends you a message in Canvas, you will see the message in the Inbox. You will also receive an email with a notice of a new message in Canvas (unless you have switched off this setting). The email also contains the message. You can reply to the message from Canvas or reply on the mail. Either way, your reply will end up in the sender’s Canvas Inbox.

Remember this when replying with an email

When you reply to a Canvas message by email, you need to remember that

you cannot see your message history in the thread, only in Canvas
you cannot send any attachment

If you want to attach a file for your recipient, you have to either reply to the message in Canvas or send the file by email.