Definition 22.25.1. Let $R$ be a ring. A *graded category $\mathcal{A}$ over $R$* is a category where every morphism set is given the structure of a graded $R$-module and where for $x, y, z \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ composition is $R$-bilinear and induces a homomorphism

of graded $R$-modules (i.e., preserving degrees).

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