Claim. A concept has two aspects — intension (qualities common to it) and extension (set of beings to which it applies) — and progressively narrowing intension yields broader extension, eventually reaching the highest generic concepts called categories.
Elaboration. Sang Hun Lee illustrates with biological classification (2. Concept). Living beings have life; animals additionally have senses; vertebrates additionally have a backbone; mammals suckle their young; primates can grasp; humans have reason. As we ascend, intension grows (more properties packed in) while extension shrinks (fewer beings included). Conversely, descending the chain (humans → primates → mammals → vertebrates → animals → living beings) shrinks intension and broadens extension.
When comparing two concepts, the one with broader intension and narrower extension is a specific (subordinate) concept; the one with narrower intension and broader extension is a generic (superordinate) concept. ‘Vertebrate’ is generic relative to ‘reptile,’ specific relative to ‘animal.’ Repeating this ascent eventually reaches the highest generic concepts — those beyond which no further abstraction is possible. These are called categories.
The term has a second sense as well: the pure a priori concepts reason possesses by nature (rather than through experience). Categories vary by philosopher because each treats his most fundamental concepts as categorial. Aristotle, taking clues from grammar, established ten: substance, quantity, quality, relation, place, time, position, condition, action, passivity. Kant established twelve based on the twelve forms of judgment.
This machinery matters for Unification logic because Lee will later propose ten primary Unification categories (Existence & Force, Sungsang & Hyungsang, Yang & Yin, etc.) derived from the structure of the Original Image rather than from grammar or judgment-forms. Categories function as the highest-level forms of thought; whose categories you adopt shows whose thought system you inhabit.
See also. judgment-aeio-and-distributed-terms, inference-deduction-induction-analogy, unification-primary-categories