The Stacks project

Definition 12.14.2. Let $\mathcal{A}$ be an abelian category. Let $A_\bullet $ be a chain complex with boundary maps $d_{A, n} : A_ n \to A_{n - 1}$. For any $k \in \mathbf{Z}$ we identify $H_{i + k}(A_\bullet ) \rightarrow H_ i(A[k]_\bullet )$ via the identification $A_{i + k} = A[k]_ i$.