Let $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ be abelian categories such that $\mathcal{A},\mathcal{B}$ have enough injectives (we can WLOG assume they're categories of modules for the following) and $G\colon\mathcal{A}\rightarrow\mathcal{B}$ and $F\colon\mathcal{B}\rightarrow\mathcal{C}$ be left-exact functors, such that $G$ sends injective objects of $\mathcal{A}$ to $F$-acyclic objects of $\mathcal{B}$. Under these conditions, there is a first-quadrant Grothendieck spectral sequence with second page $E_2^{p,q}=(R^pF)(R^qG)(A)$ (the $R^{\bullet}$ denote right-derived functors) and converging to $R^{p+q}(FG)(A)$ for any object $A\in\mathcal{A}$ (in a natural manner, though we will keep $A$ fixed). Now, the fact that the spectral sequence is first quadrant means that we have two edge maps ($F^{\bullet}$ denotes the filtration of the limit object, indexed contravariantly) $$R^q(FG)(A)\twoheadrightarrow R^q(FG)(A)/F^1R^q(FG)(A)\cong E_{\infty}^{0,q}\hookrightarrow\dotsc\hookrightarrow E_2^{0,q}=F(R^qG)(A),\\ (R^pF)(GA)=E_2^{p,0}\twoheadrightarrow\dotsc\twoheadrightarrow E_{\infty}^{p,0}\cong F^pR^p(FG)(A)\hookrightarrow R^p(FG)(A).$$ In Weibel's book, it is said that these edge maps are the "natural maps". However, I do not see any obviously describable maps between these. My questions are:
What are the natural maps? I.e. how do we describe them a priori?
How do we actually see that these (once they have been defined) agree with the edge maps from the Grothendieck spectral sequence?
It appears that Weibel doesn't really address either question and other standard texts, as such as Rotman's, seem to not discuss the edge maps at all. The closest to a reference discussing this question that I could find is this MO question, which, however, only gives partial and rather cryptic answers to 1. and does not address 2. at all. I've tried reverse engineering the edge maps from the construction of the Grothendieck spectral sequence using Cartan-Eilenberg resolutions, but the amount of natural identifications going on throughout has made me lose track at clearly identifying maps obtained at the end.
The reason I care is that the Inflation-Restriction sequence for (continuous) group cohomology can be derived from the Hochschild-Serre spectral sequence, which is a special case of the Grothendieck spectral sequence. However, while deriving an exact sequence with the right objects from this spectral sequence is not particularly hard from general spectral sequence formalism (and is done in a lot of places), the finer detail of showing that the maps obtained by this formalism are actually the inflation/restrictions maps as they are a priori defined requires knowledge of these edge maps, and I have not found any discussion of this. My hope, on the basis of Weibel calling these maps "natural", is that 1. and 2. admit reasonable answers, and the application to the Hochschild-Serre spectral sequence then being a mere formal afterthought. But, for now, I'm stuck.
Edit: I'm starting to understand the first edge map, as per the linked MO answer. Indeed, derived functors are universal among non-exact $\delta$-functors and the $F\circ R^qG$ constitute a non-exact $\delta$-functor. The a priori description of the natural maps $R^q(FG)(A)\rightarrow F(R^qG)(A)$ ought to be that these are components of the unique natural transformation of $\delta$-functors, which is the canonical isomorphism $R^0(FG)\cong FG\cong F\circ R^0G$ in degree $0$.
It remains to see why this describes the edge morphisms in the Grothendieck SS. Two facts, left as exercises in Weibel chapter 5.7, are that chain maps between complexes lift to chain maps between Cartan-Eilenberg resolutions and chain homotopies between such maps lift to chain homotopies between the Cartan-Eilenberg resolutions. Furthermore, one can check that homotopic maps between double complexes induce the same map between their spectral sequences from page $2$ onward, and also on the graded pieces of the total complexes cohomology in a compatible manner. From these observations, plus the analogous facts about injective resolutions on their own, it follows that the Grothendieck SS (taken to start at page $2$) is well-defined up to canonical isomorphism and functorial in $A$. This functoriality includes the convergence data. This implies the edge map $R^q(FG)\Rightarrow F\circ R^qG$ is a natural transformation. However, I lack the clarity to see why it is compatible with the differentials in the long sequences coming from short exact sequences in $\mathcal{A}$, i.e. why they are natural transformations of $\delta$-Functors.