There is a question completely ignored in the literature which is very strange considering how natural this question arises. I regard this as an omission and I hope to be able to clarify this matter here.
Let $R$ be a commutative ring and let $\mathcal{C}$ be the category of cochain complexes in the category of $R$-modules. Let $I^{\bullet}$ be an invective resolution of an $R$-module $M$. Is then the complex $I^{\bullet}$ an injective object in $\mathcal{C}$ in the sense that the functor $\mathrm{Hom}_{\mathcal{C}}(-, I^{\bullet})$ is an exact functor? If not then what are the injective object in the category of cochain complexes? Thanks.