Messages sync with other Apple products.

Access your messages on all your Apple devices:

Turn on Messages in iCloud

1. On your Mac, choose Apple menu  > System Preferences, then do one of the following:

• Sign in: Click Sign In, then sign in with your Apple ID. Use the same Apple ID you use to sign in to Messages.
• Verify your Apple ID: Click Apple ID , then make sure you’re signed in to iCloud with the same Apple ID you use to sign in to Messages.

2 Click Password & Security in the sidebar, then make sure that two-factor authentication is turned on. See Set up your Apple ID password and security information.

3 In the Messages app on your Mac, choose Messages > Preferences, then click iMessage.

4 In the Settings pane, select Enable Messages in iCloud.

The messages from your other devices that use the same Apple ID appear on your Mac.

Note: Depending on how many messages you have, it may take a while for all of them to appear. You can send and receive messages during that time.

To make sure you’re seeing the latest messages in iCloud on your Mac, choose Messages > Preferences, click iMessage, then click Sync Now in the Settings pane.

Actually my Mac is set up this way already.

This would have been good information to include in your original post 😀

Do you also have text message forwarding set on your iPhone? Settings > Messages > Text Message Forwarding.