Let $\mathfrak C$ be a Grothendieck category and let ${\bf D}=\mathrm{D}(\frak C)$ be its derived category, that is, consider the injective model structure on the category $\mathrm{Ch}(\frak C)$ of cochain complexes over $\frak C$ and then ${\bf D}=\mathrm{Ho}(\mathrm{Ch}(\frak C))$ is the category one obtains inverting weak equivalences.
Consider now a countable tower of objects $$\dots \to X_3\to X_2\to X_1\to X_0\ \ \ \ \ (*)$$ in ${\rm Ch}(\frak C)$. The homotopy limit $\operatorname{holim} X_i \in{\bf D}$ of this tower is defined as the limit in ${\rm Ch}(\frak C)$ of a tower of fibrant replacements. More explicitly, for all $i\in\mathbb N$ choose a quasi-isomorphism $X_i\to E_i$, where $E_i$ is a dg-injective and notice that there is an induced tower $$\dots \to E_3\to E_2\to E_1\to E_0\ \ \ \ (**)$$ By definition, $\operatorname{holim} X_i$ is the object of ${\bf D}$ represented by $\operatorname{lim} E_i$.
Since $\bf D$ is a triangulated category and it has products, we can apply another (a priori different) definition of "homotopy limit". Indeed, consider the morphism $$\phi:\prod_i X_i\to \prod_iX_i$$ where $\phi$ is the identity minus the shift map induced by the maps in the original tower. Then, the "homotopy limit" is defined as the object in $\bf D$ represented by the cone of $\phi$.
My question is: is it true that homotopy limits and "homotopy limits" are the same in this context?
I think that a positive answer for homotopy colimits is given in this paper: http://arxiv.org/pdf/1009.5904v3.pdf (appendix 1). Anyway here the authors work in the context of deviators which I'm not really familiar with. Can anybody give some insight in the particular situation I'm asking for?