Downloading webmail to Apple Mail client

Can anyone help me?
For my previous job, I was using MS Exchange email (on a web browser interface). I'd like to download my inbox and outboxes for my personal archives.

I'm using Mac Mail (on Yosemite) for the first time to do this. In my Mac Mail, I can see all of my "sent" emails going back years on my local client. However, the "inbox" is empty.

Any thoughts on how I can can spur it to download these previous inbox messages?

Thanks!

It depends on how your ISP has their server setup. For example: In gmail I can access the server from their web address and move mail from certain categories to my inbox where it can be downloaded again. Also, are you sure you want to move all your previous sent mail to your inbox as that can be overwhelming depending on much mail you've been sending out?

Thanks.

I definitely want to move all the mail, especially the inbox.

Re: the settings -- Let me put it this way, my iPhone can pull new inbox mail. BUT, my Mac Mail set up will not.

Thoughts?

Mail (the Mac program) has been able to support Exchange email accounts for quite some time. Have you set up your Exchange email account in Mail?