# Derived $p$-completion : injective in the category of diagrams?

I'm reading a set of notes that has an interlude about derived $$p$$-completions of abelian groups.

My first question stems from the fact that this interlude, although interesting, is not very furnished, so I would like to know if you knew some more references about this derived $$p$$-completion thing.

My more important question is about the definition of derived $$p$$-completion and more generally derived limit in this case. The context is the following : we have an abelian group $$A$$, its $$p$$-completion is the limit of $$A/p^k$$; we wish to make it more "homotopy friendly" in a way, so instead we take $$A//p^k$$ which is the non-negatively graded (for simplicity, perhaps "bounded below" is a better idea here, if so please tell me) chain complex $$\dots\to 0\to A\to^{p^k} A$$ which is some derived version $$A/p^k$$, and then we wish to take the derived limit of these complexes.

My question comes from the definition of derived limit in this context : we have a category $$Ch(\mathbf{Ab})_{\geq 0}$$ which has a wide subcategory $$W$$ of weak equivalences (which are the quasi-isomorphisms), and if I'm not mistaken this suffices to define what "homotopy limits" are (which I assume are what we mean by "derived limit"). It would mean a homotopical functor $$\mathbb{R}\lim : Ch(\mathbf{Ab})_{\geq 0}^\mathbb{N}\to Ch(\mathbf{Ab})_{\geq 0}$$ with a morphism $$\lim\to \mathbb{R}\lim$$ satisfying certain properties (I am using Riehl's Categorical homotopy theory for the definitions); and according to Riehl's book, such a functor is given by $$\lim\circ F$$ if $$F: Ch(\mathbf{Ab})_{\geq 0}^\mathbb{N}\to Ch(\mathbf{Ab})_{\geq 0}^\mathbb{N}$$ is a right deformation onto a subcategory on which $$\lim$$ is homotopical (we also have a transformation $$f:id\to F$$ as part of the data)

Now the notes I mentioned say that to compute $$\mathbb{R}\lim_k C(k)$$ you take a functor $$D\in Ch(\mathbf{Ab})_{\geq 0}^\mathbb{N}$$ which is injective in this category of diagrams together with a (pointwise) quasi-isomorphism $$C\to D$$ and compute $$\lim_k D(k)$$. My question is now :

why does this work ? Why is $$\lim$$ homotopical on the subcategory of injectives ? On the other hand, is it sufficient to take $$D$$ such that all $$D(k)$$'s have injective chain groups [I know that this doesn't imply that $$D$$ is injective, it's another question] ?

EDIT: I realized that maybe what was meant was to consider chain complexes of injective diagrams, not injective objects in $$Ch(\mathbf{Ab})_{\geq 0}^\mathbb{N}$$ ! That is, use the fact that $$Ch(\mathbf{Ab})_{\geq 0}^\mathbb{N}\simeq Ch(\mathbf{Ab}^\mathbb{N})_{\geq 0}$$ and take chain complexes of injective objects in $$\mathbf{Ab}^\mathbb{N}$$, which would definitely make more sense. Is that indeed what was meant ?