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

1. objects $A = (A^ i)$ are families of objects $A^ i$, $i \in \mathbf{Z}$ of objects of $\mathcal{A}$, and

2. 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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