Let $\mathcal{A}$ be a Grothendieck category (I care mostly about modules over a ring). Let $\operatorname{Ch}^+(\mathcal{A})$ the category of bounded below cochain complexes, $\operatorname{D}^+(\mathcal{A})$ its derived category. Let $\mathcal{I}$ be a (small) index category.
Consider the limit functor $\lim_\mathcal{I}$ from the functor category $\mathcal{A}^{\mathcal{I}^{op}}$ to $\mathcal{A}$. Since this is an additive functor between abelian categories with enough injectives, it has a total right derived functor $$R\lim_\mathcal{I}\colon \operatorname{D}^+(\mathcal{A}^{\mathcal{I}^{op}}) \to \operatorname{D}^+(\mathcal{A}).$$ Also note that because $\mathcal{A}$ is a Grothendieck category, $\operatorname{D}^+(\mathcal{A}^{\mathcal{I}^{op}})$ is naturally a full triangulated subcategory of $\operatorname{D}^+(\mathcal{A})^{\mathcal{I}^{op}}$.
This question has two parts.
Part 1: Is there a difference between $R\lim_\mathcal{I}$ and $\operatorname{holim}_\mathcal{I}$? If not, why the new notation?
Part 2: Is there a mapping-telescope-like representation of $\operatorname{holim}_\mathcal{I}$ (and of $R\lim_\mathcal{I}$ if they're in fact different).
