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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