This paper investigates how non-invertible symmetries and anomalies affect the crossing symmetry of S-matrices in two-dimensional quantum field theories. The study focuses on integrable flows from unitary minimal models to gapped phases, where non-invertible symmetries are preserved. The authors show that the standard four properties of S-matrices—unitarity, crossing symmetry, Yang-Baxter equation, and non-invertible symmetries—are mutually incompatible. They propose modified S-matrices that satisfy all but crossing symmetry, with crossing rules determined by fusion categories. These rules also apply to theories with discrete anomalies. The paper presents two main examples: the $\phi_{1,3}$-deformation of minimal models and the $\phi_{2,1}$-deformation of the tricritical Ising model. In both cases, the standard S-matrices derived from the bootstrap approach are incompatible with non-invertible symmetries. The authors derive new S-matrices that are consistent with these symmetries and obey modified crossing rules. They also consider the $SU(2)_1$ Wess-Zumino-Witten model with a $Z_2$ 't Hooft anomaly, showing how the modified crossing rules apply in this case. The paper concludes that crossing symmetry of S-matrices is modified in the presence of non-invertible symmetries or anomalies. The modified crossing rules are determined by fusion categories and are consistent with the TQFT dynamics in the infrared. The results suggest that non-invertible symmetries and anomalies play a fundamental role in determining the structure of S-matrices in two-dimensional quantum field theories. The study opens up new directions for research, including the exploration of modified crossing in higher dimensions and the application of categorical language to understand soft IR dynamics in gravity and gauge theory.This paper investigates how non-invertible symmetries and anomalies affect the crossing symmetry of S-matrices in two-dimensional quantum field theories. The study focuses on integrable flows from unitary minimal models to gapped phases, where non-invertible symmetries are preserved. The authors show that the standard four properties of S-matrices—unitarity, crossing symmetry, Yang-Baxter equation, and non-invertible symmetries—are mutually incompatible. They propose modified S-matrices that satisfy all but crossing symmetry, with crossing rules determined by fusion categories. These rules also apply to theories with discrete anomalies. The paper presents two main examples: the $\phi_{1,3}$-deformation of minimal models and the $\phi_{2,1}$-deformation of the tricritical Ising model. In both cases, the standard S-matrices derived from the bootstrap approach are incompatible with non-invertible symmetries. The authors derive new S-matrices that are consistent with these symmetries and obey modified crossing rules. They also consider the $SU(2)_1$ Wess-Zumino-Witten model with a $Z_2$ 't Hooft anomaly, showing how the modified crossing rules apply in this case. The paper concludes that crossing symmetry of S-matrices is modified in the presence of non-invertible symmetries or anomalies. The modified crossing rules are determined by fusion categories and are consistent with the TQFT dynamics in the infrared. The results suggest that non-invertible symmetries and anomalies play a fundamental role in determining the structure of S-matrices in two-dimensional quantum field theories. The study opens up new directions for research, including the exploration of modified crossing in higher dimensions and the application of categorical language to understand soft IR dynamics in gravity and gauge theory.