I can't find my photo library on my new iMac desktop

But you don't launch photos from your mac system preferences.

You launch photos as normal (from the photos icon in your dock, or in launchpad), and then go to the Photos>preferences menu to get the screenshots shown above.

You will find the iCloud > Photos Settings in System Preferences, when you click AppleID in the first row of System Preferences items. There you will see the iCloud settings, when you select your iCloud AppleID in the left column. Make sure, the checkmark for Photos has been set.

But the two screenshots posted by Tony are from the Photos > Preferences (in the Photos.app), where you also need to enable iCloud Photos.

To get the photos from your iPhone to your new Mac, you need first to create a new empty, Photos Library and then let the Mac sync the photos from iCloud to this new library.

• Launch Photos, and select to create a new library, when you see the message about the missing library.
• After Photos opens with the new, empty library, go to the main menu bar.
• In the main menu bar click the menu item "Photos", then "Preferences." There you will find the two tabs described in Tony's screenshot.

Hi

Have you correctly set up iCloud photos on your mac in Photos preferences? The button "use as system photo library should be greyed out, and the iCloud tab should look as shown below.

Hi - You need to be launching photos on your mac, and going to photos preferences there, not on your phone.

(Or what do you mean by "under apple ID")?

Welcome - glad it is sorted 🙂