Stuck on Software Update Complete - Loop in Iphone 13

Hi leebully,

Thanks for contacting Apple Support Communities!

We understand that the iOS update has completed the install, but it won't progress past the alert that your iPhone has been updated.

When you say it's looping, is your iPhone restarting? Have you restarted your iPhone yet? If not, we'd suggest that you restart and test again: Restart your iPhone - Apple Support

If you're still seeing the same alert, we'd like to have you connect your iPhone to your computer to complete the update as outlined here: If you can't update or restore your iPhone or iPod touch - Apple Support

"Put your device in recovery mode and set it up again

1. Make sure that your Mac is up to date. If you're using iTunes, make sure you have the latest version of iTunes. 
2. On a Mac with macOS Catalina or later, open Finder. On a Mac with macOS Mojave or earlier, or on a PC, open iTunes. If iTunes is already open, close it, then open it again. Find out which macOS your Mac is using.
3. Keep your device connected and wait until the recovery mode screen appears:

• iPhone 8 or later, including iPhone SE (2nd generation): Press and quickly release the volume up button. Press and quickly release the volume down button. Then, press and hold the side button until you see the recovery mode screen.

4. Locate your device on your computer. When you see the option to Restore or Update, choose Update. Your computer will try to reinstall the software without erasing your data. Wait while your computer downloads the software for your device. If the download takes more than 15 minutes and your device exits the recovery mode screen, let the download finish, then repeat step 3.

5. After the Update or Restore completes, set up your device."

We hope this helps.

Cheers!