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)
