Quick clarification on how the essay operationalizes the word 'cardinal': it's really meant to refer to something which the sensibles-concepts, which we receive thanks to our conceptual capacities, unify into. It's the object of recognition in Travis's terminology. It's the very first general concept that the instance object falls under. In McDowell's terminology, it's the concept that we construct by 'isolating' the specific 'aspect' of the intuition 'in speech or judgment' (using language), in hope to best 'capture' the 'given' 'intuitional unity' of the sensibles in the intuition. So, it's more constrained by the intuition -- product of sensibles-concepts' unification and designed to capture the intuitional unity -- compared to other concepts -- greater than, equal to, rational preference, etc. -- which are not given in the intuition and which we thus apply more freely based on our network of conceptual capacities.
Here is the essay! So What Exactly is a ‘Tree’ Again?