Restart to update to Catalina 10.15.7 is restarting without updating

First, open About This Mac from the Apple menu. Verify the version of macOS that is installed. To update to 10.15.7, you must have 10.15.6 already installed. To update from any prior version of .Catalina, you need to use the 10.15.7 Combo Updater - Download and Install the macOS Catalina 10.15.7 Combo Update. I recommend always using Combo updaters because they cause few problems.

Second, use Disk Utility to run First Aid on your startup volume:

Repair the Drive for El Capitan or Later

1. Restart the computer. Immediately, at or before the chime, hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
2. Select Disk Utility and press the Continue button.
3. Then select the indented (usually, Macintosh HD) volume entry from the Device list. 
4. Click on the First Aid button in the toolbar and wait for the Done button to activate. Click on it, then quit Disk Utility.
5. Select Restart from the Apple menu.

Third, look at this check list:

.Before You Upgrade macOS

1. Fix any problems before upgrading.
2. Run First Aid in Disk Utility while booted from your Recovery HD.
3. Backup. Make a bootable clone using CarbonCopy Cloner or use TimeMachine.
4. Be sure your startup disk has 18-20GB of free space during the installation. More space is better.
5. Upgrade third-party software for compatibility. See App Compatibility Table - RoaringApps. If you have mission critical software, then DO NOT upgrade until you know that software is compatible with the version of macOS you are about to install.
6. Disconnect all third-party peripherals.
7. Download the installer which will be in the Applications folder. Quit the installer after it opens, then make a copy of the installer and save it in the Downloads folder. The one in the Applications folder will be deleted after a successful installation.
8. The download is large, anywhere between 8-9GBs. It takes time to complete. Be patient.
9. Double-click on the installer in the Applications folder. Click on the start button to begin the installation.

If you are currently running an earlier version of Catalina, then updating requires using Software Updates preferences. You may also try Downloading and Installing the macOS Catalina 10.15.7 Combo Update.

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