If we go a step back for a moment... If we shoot a target 2 times, once with a light source filtered to 5000K, and once with a light source filtered to 6500K, we will have 2 sets of device values that are not in linear proportion to each other. Same, if we take 2 shots under daylight sunny and daylight overcast. The usual point here is that the results should not look the same anyway due to human perception. However it is a general statement, applied backwards, to justify already existing math. Contrary to scanners where the light spectrum is fairly stable, and viewing prints under D50 lights in booths, photography deals with a much more diversity in light sources, and constant complaints on wrong colour is not because photographers and clients are being capricious, but because there is a problem there indeed. This problem stems mostly from the nature of CATs.
For now, the only publicly available way to tame CATs in wrong places is to create ad hoc profiles, that is to profile for the light is the scene (easy in studio, very difficult anywhere on location). What happens when an ad hoc profile is created usually is that the CAT is an organic result of computing a transform between a shot taken under studio lights and target reference being D50/2°. Effectively it is what we here refer to as "re-lightning". But in this case we try to use not "some" lights, but lights as close to D50 as possible - and that helps to keep adaptation to minimum. Try the same with an arbitrary light - and the results are not usable. This alone demonstrates the problem we are having with CATs, "re-lightning", and profiling.