Functorial injective resolution out of functorial injective embedding Let $\mathsf A$ be a Grothendieck abelian category. The category of positive cochain complexes in $\mathsf A$ is also Grothendieck abelian.
One can construct an endofunctor on cochain complexes $\bf I$ along with a natural transformation $i:1\Rightarrow \bf I$ which is pointwise monic and such that $\mathbf I(C^\bullet)$ is injective.
Is there a way to produce out of such a functorial injective embedding a functorial injective resolution? That is, a functor a natural transformation $r:1\Rightarrow R$ such that $R(C^\bullet)$ is injective and $r$ is componentwise a quasi-isomorphism?
Maybe this is classical homological algebra, but I'm a bit lost. Note we begin with a complex, not just an object. Perhaps some inductive procedure mimicking the case of an object?
 A: I believe the answer is yes. 
I won't use the fact that the category is Grothendieck abelian, and so I may dualize the problem to projective resolutions (I'll do this essentially because I'm more used to homology than cohomology, but also because I'll use the Dold-Kan correspondance, which I know stated in homological notation). The constructions I'll do will be dualizable. 
EDIT : I misunderstood part of the question, but it's not a problem. To understand what I mean, consider that $\mathbf I$ is only defined on objects of the abelian category, not on cochain complexes. Then see my edit completely below to see how to get from there to a functorial injective resolution of objects.
The construction is as follows : take a (nonnegative, homologically graded) chain complex $C_*$. By Dold-Kan, we may view it as a simplicial abelian group $\mathcal C_\bullet$. Then apply $\mathbf P$ (the dual of what you called $\mathbf I$) dimensionwise to get a simplicial chain complex $\mathbf P(\mathcal C_\bullet)$, together with a map $\mathbf P(\mathcal C_\bullet)\to \mathcal C_\bullet$ which is a dimension-wise projective resolution (where I'm including simplicial abelian groups into simplicial chain complexes in the obvious way). 
Then take the total complex of that : in degree $n$ it has $K_n = \bigoplus_{p+q=n}\mathbf P_p(\mathcal C_q)$, and the double complex of $\mathcal C_\bullet$ is simply the obvious chain complex associated to $\mathcal C_\bullet$, which is known to be (functorially) quasi-isomorphic to $C_*$, in fact there's a morphism of chain complexes from this one to $C_*$ which is a (natural) chain homotopy equivalence, so we may as well think of $\mathcal C_\bullet$.
So we get a natural chain map $K_*\to \mathcal C_*$ ($\to C_*$). It remains to check that this is a projective resolution. Clearly each $K_n$ is projective (direct sum of projectives), and moreover we have a spectral sequence with $E^1_{p,q} = H_q(\mathbf P_*(\mathcal C_p))$ which converges to the homology of the total complex. This is natural, so we also get one for $\mathcal C$, and the induced morphisms are the correct ones. The point is that $E^1_{p,q} = \mathcal C_p$ if $q=0, 0$ if $q>0$ (by definition of projective resolution), so actually $K_*\to\mathcal C_*$ is a quasi-isomorphism; and so $K_*\to C_*$ is one as well. 
This is clearly a functor (we defined it as a composite of functors), and the transformation si clearly natural, so this is what we want. 
I'm sort of too lazy to write down the dualization, but in any case you can see that I only used finite coproducts so no hypothesis of the form "Grothendieck", I really only used abelian-ness, and since $Ch_{\geq 0}(A^{op})^{op} = Ch^{\geq 0}(A)$ for any abelian category $A$, we get what we want with injective resolutions.
Note that the use of Dold-Kan is essential here, because if you apply $\mathbf P$ to $C_*$, you get something which is no longer a chain complex (indeed $\mathbf P$ is not additive most of the time, and so $\mathbf P(0) \neq 0$ in general; the best example being in abelian groups, where you can take, e.g. $\mathbf P_0$ to be $\mathbb Z[-]$)
See here for information about the Dold-Kan correspondance, and here for the spectral sequence I'm using (although I think they describe it for the other filtration, but it's not different)
EDIT : I noticed I didn't explain how to get a functorial projective resolution from a functorial projective epimorphism - you probably know that but let me spell it out anyway : define a projective resolution by take a projective epimorphism $\epsilon : P_0\to X$, then a projective epimorphism $d_1 : P_1\to \ker \epsilon$, then a projective epimorphism $d_2 :P_2\to \ker(d_1)$, etc. This is clearly functorial if you are given a functorial projective epimorphism. 
A few additional notes : This total complex construction can be made to work to build projective resolutions of complexes out of projective resolutions of objects, but if you're doing that "by hand", it won't be functorial - and you probably can't use Dold-Kan so easily. The idea is that if you have your complex $C_*$, you resolve $C_0$ by $P^0_*$, more generally $C_n$ by $P^n_*$, and dimension-by-dimension you lift $d_n :C_n\to C_{n-1}$ to a (unique up to chain homotopy) chain map $P^n_*\to P^{n-1}_*$. This gets you a double complex, which can again be totalized to a projective resolution of $C_*$. 
The nonfunctoriality comes from the fact that the chain maps between the $P^n_*$'s you get are determined only up to homotopy, and not strictly. The use of Dold-Kan in the above proof is there to ensure functoriality.
Note also (if you have some background in model category theory) that in nice enough situations, you can use the small object argument to get functorial cofibrant replacements in the projective model structure; and those correspond to projective resolutions. I don't know how well this dualizes, though (in the projective case you'll need something like compact projective generators in your abelian category, so dually you'd need something like cocompact injective cogenerators, and I'm not sure how often these exist)
