Restore a repaired iPhone 5 from backup
You have to make a backup of the loaner phone to have it available to restore on your repaired phone.

You already appear to have done so on your phone, but if needed, steps described by Apple are:


  Make a backup using iTunes
  
  
  Open iTunes and connect your device to your computer. 
  Save content that you downloaded from the iTunes Store or App Store by clicking File > Devices > Transfer Purchases. If you're using Windows and you don't see a File menu, press the Alt key.
  
  
  
  
  
  Wait for iTunes to finish transferring your purchases. Then, click File > Devices > Back up. If you're using Windows and you don't see a File menu, press the Alt key.
  
  
  
  
  
  After the process ends, you can see if your backup finished successfully by clicking iTunes Preferences > Devices. You should see the name of your device with the date and time that iTunes created the backup.
  
  
  
  
  Follow more steps if you need help making an iTunes backup. Learn how to encrypt your backup.