Grothendieck spectral sequence

Take the supplied injective resolution $A\to I$ and a Cartan–Eilenberg resolution $F(I)\to J$, or obtain the latter by F1. Apply $G$ to $J$ and use the filtration by its resolution degree. F2 gives $E_2^{p,q}=R^{p}G(R^{q}F(A))$, identifies its total target with $R^{p+q}(GF)(A)$, and identifies both edge maps. All indices are nonnegative.

The finite total diagonals of $GJ$ give exactly the finite target filtration in F2: in degree $n$ only resolution degrees $0,\dots ,n$ occur. The associated graded is its stationary page, with differential bidegree $(r,1-r)$, so this is strong convergence, not just an asserted target. In degree zero the sole quotient is $GF(A)$; zero objects and zero filtration pieces require no separate reconstruction.

For $A\to A^{\mathrm{\prime }}$ lift to $I\to I^{\mathrm{\prime }}$ using F3 or supplied data, apply $F$, then lift to $J\to J^{\mathrm{\prime }}$ by F4. This preserves resolution degree and gives a spectral-sequence map. On $E_2$ it is the map $R^{p}G(R^{q}F(A))\to R^{p}G(R^{q}F(A^{\mathrm{\prime }}))$. Different injective comparison maps are homotopic, so additivity of $F$ gives the same maps on $R^{q}F$, and hence on $E_2$; different Cartan–Eilenberg lifts give the same $E_2$ map by F4. Equality propagates to every later page by taking homology. On the target, additivity of $GF$ preserves the original injective homotopy and the augmentation comparison in F2 intertwines the maps, so the target maps also agree. Identity and composition now establish naturality.