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

$\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(y, z) \otimes _ R \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, z)$

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

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