Update

That refers to your startup drive, not the download. Does the computer have a HDD or an SSD? If First Aid says the disk is damaged,, then the first thing to try is to erase the drive and install macOS. You can try this:

Install El Capitan or Later from Scratch

If possible, back up your files.

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 from the Utility Menu and click on the Continue button.
3. When Disk Utility loads select the target volume (indented entry, usually named "Macintosh HD") from the Device list.
4. Click on the Erase button in Disk Utility's toolbar. A panel will drop down.
5. Set the Format type to APFS (SSDs, only for Mojave and Catalina) or Mac OS Extended (Journaled.)
6. Click on the Apply button, then wait for the Done button to activate and click on it.
7. Quit Disk Utility and return to the Utility Menu.
8. Select Install OS X and click on the Continue button.

Throw your current download into the Trash, then try again. Be sure you have a fast Internet connection. I recommend using Ethernet rather than Wi-Fi, if you can. Be sure you run First Aid on your startup volume before trying to re-download Catalina.