= Solution
\b[Theorems matter at least as much as definitions: they show when a categorical description captures actual mathematical structure.] Definitions in <category theory> are exceptionally economical, but economy alone does not explain their applications.
An <adjunction> compresses a universal mapping property into natural hom-set bijections. The free-group <adjunction>, for example, records that assigning images to generators extends uniquely to a homomorphism. That is a useful definition, but proving that a free group exists and satisfies this property still requires mathematics. The definition organizes that proof rather than replaces it.
The <Yoneda lemma> offers a sharper example. A <representable functor> is easy to define; the theorem identifying <natural transformations> out of it with elements of another <functor> gives the definition its computational force. The representation of a set-valued <functor> as a <colimit> of representables is then a substantive consequence, and the theorem that <filtered colimits> commute with <finite limits> turns it into a criterion for finite-limit preservation. None of these conclusions follows merely from choosing attractive terminology.
Likewise a <monad> packages operations and their compatibility, while the <Eilenberg-Moore category> defines the corresponding algebras. The <Beck monadicity theorem> answers a different question: whether a given <category> is really recovered by that algebraic description. Applying it to <compact Hausdorff spaces> needs a topological argument about closed quotient relations. The definitions would not by themselves predict or justify the result.
Finally, <abelian categories> put <kernels in a category>, <cokernels in a category> and exactness into a setting where homological arguments can be reused. Some consequences, such as image factorization, become short because strong structural content has deliberately been included in the definition. Yet <left derived functors>, long exact sequences and the detection of projective objects supply information about obstructions to exactness which the terminology alone cannot provide.
Definitions and theorems therefore play complementary roles. A good definition states what should be invariant across examples; a theorem proves which constructions preserve it, which examples satisfy it, and what can be deduced from it. <Category theory> makes that partnership unusually visible, rather than making proofs less important.
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