Derived composition isomorphisms under total acyclicity

Under the first condition all rows $q>0$ vanish, leaving $E_2^{p,0}=R^{p}G(F(A))$. Under the second condition F2 makes every column $p>0$ vanish, leaving $E_2^{0,q}=G(R^{q}F(A))$. The qualification including $q=0$ is needed to kill entries $E_2^{p,0}$ with $p>0$. In either case every differential $d_r$ for $r\ge 2$ has zero source or target because it changes both coordinates. The same support is preserved on taking homology, hence $E_2=E_{\mathrm{\infty }}$.

The first case has one possible degree-$n$ quotient at filtration index $n$; the zero preceding quotients identify its filtration subobject with all of $H^n$. The second case has its sole quotient at index zero; the zero later quotients force $F^1{H}^n=0$. The normalized endpoints in F1 therefore identify the respective edges with the displayed isomorphisms. At $n=0$ both reduce to $GF(A)$, and if the sole quotient is zero the target is zero by the same finite argument. Naturality follows by restricting F1 to morphisms between inputs obeying the conditions.