The traditional way to use Lightroom Classic on multiple computers is to keep your image files and your catalog files on a rugged external drive. Then move it from computer to computer. It used to be felt that doing that slowed down Lightroom Classic. But, if you use the newer, faster (more expensive!) solid state drives, speeds are better.

Adobe's promotions will insist that their preferred multi computer solution is Lightroom (the cloud model). There are versions of that for computers, phones, tablets and browsers and all will stay synchronized.

As Paul explains, you can have "shared collections" out of Lightroom Classic that sync with Lightroom (cloudy). I do some of that for a few favorite images that I want to link further to the Adobe "Portfolio". Syncing can be slow, especially if you are using RAW files. But, depending on your workflow, smaller RAW previews are what get can be "synced".

I've never seen, in Adobe's forums or elsewhere, the method Paul describes where you have two computers running Lightroom Classic and use Lightroom (cloudy) as a "bridge" between smart, synced collections. It sounds like it might work, but it might take more time getting it to work reliably than is worth it.

For reference, Adobe's instructions explain sharing from Lightroom Classic to Lightroom (cloudy): https://helpx.adobe.com/lightroom-classic/help/share-comment-feedback-collaborate.html