Definition 12.16.1. Let \mathcal{A} be an additive category. The category of graded objects of \mathcal{A}, denoted \text{Gr}(\mathcal{A}), is the category with
objects A = (A^ i) are families of objects A^ i, i \in \mathbf{Z} of objects of \mathcal{A}, and
morphisms f : A = (A^ i) \to B = (B^ i) are families of morphisms f^ i : A^ i \to B^ i of \mathcal{A}.
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