Depending on how often you switch between computers, storing it on an external hard drive can be the best solution. Backing it up to the NAS would make sense. It can be a bit annoying to drag around the hard drive, but in some cases it makes sense.
Again, depending on your needs, another possibility is to store the main catalogue (always have a "main" catalogue) on your main computer and then rely on LR's sync feature to give the laptop access to the images and work done on them. So you might have the 580k catalogued in the file on your desktop PC and sync some files to Adobe (eg recent ones, ones you're working on, even everything). You can then access them in LrMobile on your phone, or on the laptop using a browser (LrWeb) or by installing Cloudy Lightroom, and do certain work (adjust, title, caption, rate and flag, but not keywords) which will sync automatically back to the main catalogue. New photos might be imported into either computer, but sometimes you might want to import them into your laptop's Classic catalogue (which you can backup to the NAS), work on them, and finally get them over to the main catalogue which can be done via a few methods depending on circumstances like quantity or network speed.
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