Syncing catalog with Adobe Cloud storage

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How do I know if my catalog is getting backed up to my Adobe Cloud storage?  Does it?

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As in the above reply. your Catalog file is stored on your computer internal drive. It does NOT get backed up to the Adobe Cloud system.

Also please note that the LrC Catalog does not hold any images. It is a Database file. All your images are stored on you computer hard drive.

Look here for tutorials on what LrC is, How it works and How to work with LrC.

Lightroom Classic tutorials | Learn how to use Lightroom Classic

Sorry but I think you're getting in the "Adobe names everything with the same name" trap. 

If you are mostly using LR-Classic than you probably do not need extra storage space. If you are using LR (both mobile and Desktop) as your primary LR engine, than you very well may need extra storage space. 

Your LR-Classic catalog is not being stored in the clouds, it's being stored on your hard drive (and I do hope you have a backup for security). 

FWIW, my primary LR is Classic but I do take advantage of LR so that when I take any photos with my phone, the images will show up (on their own, I need to do nothing) in LR-Desktop. In addition, if I move selected images from LR-Classic into a Collection, and I sync that Collection to the Clouds, I can share those images with friends from my phone or iPad. But those are not original images so do not think of them as backup. Also, these synced images will show up in LR-Web so I can send a link to friends to see on their own.

Suffice it to say, I do not know how you are using LR or LRC so I can't really advise you. However, if you do start up LR-Desktop you can click on the Cloud icon (far right corner of the LR-Desktop) and it will show you how much of your storage you are using. If you do not expect to be using as much as you're paying for, you'll have to call Adobe and adjust your rates. I do not work for Adobe, and it's very likely that no one from Adobe will see this message here so do not expect anything to happen from here. 

Good luck!

Lightroom Classic does not store your original image files, your Catalog, Preview files, Backups in your Cloud Storage. That is all managed on your localized computer internal, external or network disk drives. Your daily backups of your Catalog file which is a data storage file and does not contain image files or copies of them. You need to make your own plans for backing up your original image files.

There is a new Lightroom application for your Computer referred to as Desktop, Cloudy or Cloud-based, see the link for more info.

https://www.lightroomqueen.com/lightroom-cc-vs-classic-features/