Why won’t my iPhone 13 Pro update to iOS 16

Before any iOS / iPadOS update, please do the following

• You need to Identify your iPhone model - Apple Support
• It is also recommended that you either take a backup on iTunes or iCloud --> How to back up your iPhone, iPad, and iPod touch

VPN or proxy connections might prevent your device from contacting the update servers.

• Get help with over-the-air iOS updates - Apple Support
• Remove and download the iOS update again

If you still can't install the latest version of iOS, try downloading the update again:

1. Go to Settings > General > [Device name] Storage.
2. Find the iOS update in the list of apps.
3. Tap the iOS update, then tap Delete Update.
4. Go to Settings > General > Software Update and download the latest iOS update.

If you don’t see the update in the list of apps or if the issue happens again, you need to update your device with iTunes.

Try updating iOS after each step.

• Try to Forget the WiFi network and add it again
• Try another WiFi network 
• Toggle sets automatically off and then on
• Reset your network settings
• Force restart your device
• Remove any beta profiles, if applicable
• If everything fails: - Use iTunes or Finder (for macOS Catalina+) to update your device instead of over-the-air
• Get help with iOS update and restore errors - Apple Support
• If you see an error when you update or restore your iPhone, iPad

Do you have a beta version of 15.7 installed?

How to uninstall iOS or iPadOS beta software - Apple Support

Then follow the instructions in the link I posted to fix the issue.