Natural morphism appearing in Grothendieck spectral sequence

Question

Assume we are in the setting of the Grothendieck spectral sequence (Weibel, 5.8): $G : A \to B, F : B \to C$ are left exact functors such that $G$ sends injective objects to $F$-acyclic objects. Now the edge maps should be some natural maps $(R^p F)(GA) \to R^p(FG)(A)$ and $R^q(FG)(A) \to F((R^q G)(A))$. But how are they defined?

I've managed to write down a definition in the special case of the Leray spectral sequence, there $G : Sh(X) \to Sh(Y)$ is a direct image functor of a map $f : X \to Y$ and $F : Sh(Y) \to Ab$ is a global section functor: The first map is $H^q(Y,f_* A) \to H^q(X,A)$ and the second map is $H^q(X,A) \to H^0(Y,(R^q f_*) A)$. For the first one, you may use the inverse image functor and define $H^q(Y,-) \to H^q(X,f^{-1} -)$ using universal $\delta$-functors, and then compose this with the adjunction morphism $f^{-1} f_* A \to A$. For the second one, you may use an injective resolution $I^*$ of $A$ and use the canonical maps $H^0(Y,Z)/H^0(Y,B) \to H^0(Y,Z/B)$ for a subsheaf $B \subseteq Z$.

But this method does not generalize.

Also, I want to know why these maps are natural with respect to $F$ and $G$. For example in the special case above, if $f' : X' \to Y$ is another map and $g : X \to X'$ is a map over $Y$ and $A'$ is a sheaf on $X'$, why is the diagram [Feel free to edit this!]

$H^q(X',A') \to H^q(X,g^{-1} A')$

$\downarrow ~~~~~~~~~~~~~~~~~~~~~~~~~~ \downarrow$

$H^0(Y,(R^q f'_*) A') \to H^0(Y,(R^q f_*) g^{-1} A')$

commutative? Motivation: This is needed to show that the morphism $Pic(X) \to Pic(X/Y)$ is natural, where $f$ is a morphism of locally ringed spaces.

Answer 1

The second edge morphism $\kappa_2: R^p(FG)(X) \to FR^pG(X)$ is induced by universality of $R^*(FG)$ from the identity on $FG(X)$.

The first edge morphism $\kappa_1: R^pF(GX) \to R^p(FG)(X)$ can be described as follows: Assume $G$ has an exact adjoint functor $T$. Choose an injective resolution $X[0] \to J^\bullet$ in $\mathcal{A}$. Apply $G$ and choose an injective resolution $GX[0] \to I^\bullet$ in $\mathcal{B}$. Apply $T$ and use the composition of the natural map $TGX[0] \to X[0] \to J^\bullet$ and the theorem on the natural extending of morphisms between an exact complex and an injective complex to obtain a morphism $TI^\bullet \to J^\bullet$. Adjoint to this is $\phi: I^\bullet \to GJ^\bullet$, which gives us $\kappa_1$ after applying $F$ and taking cohomology. (see also commutative diagram with Yoneda pairing, Weil pairing and edge morphism)