THE SZYMCZAK FUNCTOR ON THE CATEGORY OF FINITE SETS AND FINITE RELATIONS

The paper proves the two target statements precisely as Theorems 2.1 and 2.2: every finite relation is Szymczak‑isomorphic to a canonical object (Theorem 6.27) and, among canonical objects, Szymczak‑isomorphism coincides with Endo(FRel) isomorphism (Theorem 6.36) . The canonicalization in the paper uses the quotient X̄ := XR/∼R with the relation R̄ induced by R, together with explicit morphisms [S,p] and [T,p] witnessing the isomorphism in Szym(FRel) . By contrast, the model’s “phase‑lift” to XR × Z/pZ with R̄(y,i) = (y,i+1) discards all inter‑component edges and would collapse distinct Szym‑classes; this contradicts the paper’s invariants (classifying graphs) which separate canonical objects that share the same global period p . Moreover, for the rigidity theorem, the paper builds new component‑wise bijective morphisms U,V using several technical lemmas (notably Lemmas 6.34–6.35) to obtain an Endo‑isomorphism, rather than claiming the original Szym‑morphisms are already inverses in Endo(FRel) . Hence the paper’s results are correct; the model’s solution makes over‑strong and incorrect reductions.