Suppose $A,B,C,D,Z$ are abelian categories. Let $G:C\longrightarrow D$, $F:D\longrightarrow Z$, $P:C\longrightarrow A$, $G':A\longrightarrow B$, $P':D\longrightarrow B$ and $F':B\longrightarrow Z$ are covariant left exact additive functors such that $P'G=G'P$, $F'P'=F$, $G$ takes injectives to $F$ acyclics, $G'$ takes injectives to $F'$ acyclics, $P$ takes injectives to $F'G'$ acyclics and $P'$ takes injectives to $F'$ acyclics. Let $q\geq 0$, $X\in \mathrm{ob}(C)$.
Then the question is whether $E_{1}E_{2}=E'_{1}E'_{2}$, where $E_{1}:R^{q}F(GX)\longrightarrow R^{q}(FG)(X)$ is the edge homomorphism when the Grothendieck spectral Sequence (GSS) is applied to the composition $C\overset{G}\longrightarrow D\overset{F}\longrightarrow Z$,$E_{2}:R^{q}F'((P'G)(X))\longrightarrow R^{q}F(GX)$ is the edge homomorphism when the GSS is applied to the composition $D\overset{P'}\longrightarrow B\overset{F'}\longrightarrow F$, $E'_{1}:R^{q}(F'G')(PX)\longrightarrow R^{q}(FG)(X)$ is the edge homomorphism when the GSS is applied to the composition $C\overset{P}\longrightarrow A\overset{F'G'}\longrightarrow Z$,$E'_{2}:R^{q}F'((P'G)(X))\longrightarrow R^{q}(F'G')(PX)$ is the edge homomorphism when the GSS is applied to the composition $A\overset{G'}\longrightarrow B\overset{F'}\longrightarrow Z$.
