Well my question is as clear as its title suggests. So here I would like to clarify on the fact that an object $A^\cdot$ in $\mathrm{Ch}^{\geq 0}(\mathcal{A})$ is injective if and only if $0\longrightarrow\mathrm{H}^0(A^\cdot)\longrightarrow A^0\longrightarrow A^1\longrightarrow\cdot\cdot\cdot$ is exact and $\mathrm{H}^0(A^\cdot)$ and all $A^n$'s are injectives in $\mathcal{A}$.

Many thanks in advance!