The paper explores the interplay between non-invertible symmetries, anomalies, and S-matrices in two-dimensional integrable theories. It finds that crossing symmetry of S-matrices is modified when the IR phase is governed by a non-trivial topological quantum field theory (TQFT). Specifically, the authors show that S-matrices derived from the bootstrap approach are incompatible with non-invertible symmetries along RG flows to gapped phases. They propose consistent alternatives that violate standard crossing symmetry but obey modified rules dictated by fusion categories. These rules extend to theories with discrete anomalies, as demonstrated in the example of the perturbed $SU(2)_1$ Wess-Zumino-Witten (WZW) model. The paper also discusses the Ward identity for non-invertible symmetries and provides a physical explanation for the modified crossing symmetry. Future directions include testing the proposed S-matrices through physical observables, clarifying the relationship between 't Hooft anomalies and modified crossing, and exploring higher-dimensional examples.The paper explores the interplay between non-invertible symmetries, anomalies, and S-matrices in two-dimensional integrable theories. It finds that crossing symmetry of S-matrices is modified when the IR phase is governed by a non-trivial topological quantum field theory (TQFT). Specifically, the authors show that S-matrices derived from the bootstrap approach are incompatible with non-invertible symmetries along RG flows to gapped phases. They propose consistent alternatives that violate standard crossing symmetry but obey modified rules dictated by fusion categories. These rules extend to theories with discrete anomalies, as demonstrated in the example of the perturbed $SU(2)_1$ Wess-Zumino-Witten (WZW) model. The paper also discusses the Ward identity for non-invertible symmetries and provides a physical explanation for the modified crossing symmetry. Future directions include testing the proposed S-matrices through physical observables, clarifying the relationship between 't Hooft anomalies and modified crossing, and exploring higher-dimensional examples.