Let $G$ be a finite group. In this paper, we study $G$-categories equipped with an ordinary symmetric monoidal structure, together with a set of specified norm maps. We give an example and explain how the Hill-Hopkins-Ravenel norm functors arise from it, and then we generalize the Kelly-Mac Lane coherence theorem to include the present structures. As an application, we obtain finite presentations of $N_\infty$-$G$-categories for any $G$-indexing system. We also prove coherence theorems for normed symmetric monoidal functors and natural transformations, and we show that normed symmetric monoidal categories are essentially determined by the indexing systems generated by their sets of norms.