Just fitting a cat doesn't bring the emissions into that range for the ULEZ.

And the range for MOT and ULEZ are different, otherwise every car on the road would be able to pass through the ULEZ, wouldn't they?

The ULEZ is a stricter set of measures for the given area, and that means that some older cars - even those with cats - won't be able to pass under that bar.

Same way that for years diesels would have passed, but since we realised what their ACTUAL emissions are, they don't qualify.

It's to do with actual recorded emissions, not age or technologies involved.