LR 2023 on same computer running LR 6 and CS6

Yes, "your cs6 licenses may be taken over by cc."- and your Lr6 licence will be taken over by CC Desktop App and become a 'subscribed' version 'LR CC2015' (30day trial).

If you subscribe to the Photography Plan 20GB with Lightroom-Classic and Photoshop (and the 'Cloud based' Lightroom) they will replace your old apps LR6 & CS6-Photoshop.

Lr-Classic v13.5 will simply make a copy of your existing LR6 catalog and update it to work with LrC 13.5, then you carry on with Lightroom-Classic as usual.(v13.5 catalogs in LrC are not backwards compatible with the old LR6. Also LR6 installs can no longer be activated. LR6 is a "dead duck")

You should always be backing up the Lr catalog to another disk- not just at times like upgrading.

in the future, to find the best place to post your message, use the list here, https://community.adobe.com/

p.s. i don't think the adobe website, and forums in particular, are easy to navigate, so don't spend a lot of time searching that forum list. do your best and we'll move the post (like this one has already been moved) if it helps you get responses.

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yes, but your cs6 licenses may be taken over by cc.