Functoriality Forces Uniformity: Sections of the Object Functor and the Structure of Sym-Uniform Categories

We study sections of the object functor Ob: Cat_sm → Set from small categories to sets. Our first result is the Symmetry Rigidity Theorem: any functorial section s must produce, for each set X with |X| ≥ 2, a category s(X) on which Sym(X) acts as automorphisms, yielding isomorphic diagonal hom-sets, isomorphic non-diagonal hom-sets, and Sym(X)-equivariant composition (Theorem 3.1). No functorial section produces any small category whose non-diagonal hom-sets vary up to isomorphism. Our second result is an algebraic classification: for |X| ≥ 4, Sym(X)-uniform categories on X, equipped with the canonical permutation action, are classified up to equivariant isomorphism by uniform category data — a monoid M_d, a set M_nd, and five composition maps subject to fifteen explicit axioms (Theorem 5.7). A completeness lemma (Lemma 5.3) verifies that the fifteen axioms correspond bijectively to the fifteen coincidence patterns of composable triples. The axioms have non-trivial content: the diamond condition forces |M_nd| ≤ 1 whenever M_d is trivial, so Disc and Indisc are the only uniform categories with trivial endomorphism monoid (Proposition 5.11). We prove the adjoint sections Disc and Indisc are initial and terminal in Cat_X (§6), identify necessary cross-cardinality coherence constraints on global sections (§4), and give a geometric interpretation via nerves: for connected groupoids, the non-recoverable content is detected by holonomy (π₁ ≅ Aut, Theorem 7.4). A philosophical application of the Symmetry Rigidity Theorem to mathematical structuralism is developed in "Objects Do Not Determine Structure" (Badkur and Dak, 2026, in preparation).