Categorical foundations of discrete dynamical systems

The paper’s Theorem 2.3 establishes the n-cycle decomposition for a skew/semidirect product (X×Y, f ⋊p g) and the independence (up to isomorphism) of the fiber system from the orbit representative, via a categorical pullback-and-slice argument in Set^{Z/n} together with limit-preservation of S, A, and ev_n and Proposition 2.2 (pullback square for semidirect products) . The candidate solution proves the same result by an explicit, elementary, Z/n–equivariant bijection, constructing inverse maps F_[c], G_[c] on each X-component orbit and checking conjugacy when changing representatives. Both are correct and yield the same isomorphism; the proofs differ in style (categorical vs. explicit combinatorial). The candidate implicitly relies on standard facts also used in the paper: cycles as maps (Proposition 1.28) and the factorization/rotation calculus on cycle graphs (Proposition 1.17, Theorem 1.32) .