The Categorical / Structural Perspective: Invariants of Structure-Preserving Morphisms

We develop the structural answer to "what is a number?" by treating numerals as invariants under all structure-preserving morphisms between models of arithmetic. We survey mathematical structuralism (Resnik, Shapiro, Hellman, Awodey), present its categorical formalization through ETCS and functorial semantics, and prove an isomorphism-invariance theorem. Univalence makes the meta-condition "a property is structural iff preserved under isomorphism" an internal theorem rather than an external rule.