Naturality of the Grothendieck spectral sequence

Fix an injective resolution $I$ of the input. Naturality of $\alpha$ makes ${\alpha }_I:F(I)\to F^{\mathrm{\prime }}(I)$ a cochain map. Lift it by F2 to $a:J\to J^{\mathrm{\prime }}$ between their supplied Cartan–Eilenberg resolutions. Apply $G$ and then $\beta$ to form the bicomplex map $GJ\stackrel{G(a)}{\to }G{J}^{\mathrm{\prime }}\stackrel{{\beta }_{J^{\mathrm{\prime }}}}{\to }{G}^{\mathrm{\prime }}{J}^{\mathrm{\prime }}$. It preserves both bidegrees, hence the resolution-degree filtration.

Taking horizontal cohomology identifies the first factor with $G$ applied to the map of the injective resolutions of $R^{q}F(A)\to R^q{F}^{\mathrm{\prime }}(A)$. Taking vertical cohomology then gives $R^{p}G$ of this map followed by the transformation $R^{p}G\to R^p{G}^{\mathrm{\prime }}$ induced by $\beta$ on the same injective resolution. This is the stated $E_2$ map. F2 makes it independent of the lift, and page homology propagates this independence to later pages.

The augmentation square from $GF(I)\to \mathrm{Tot}(GJ)$ to $G^{\mathrm{\prime }}{F}^{\mathrm{\prime }}(I)\to \mathrm{Tot}(G^{\mathrm{\prime }}{J}^{\mathrm{\prime }})$ commutes: $a$ extends ${\alpha }_I$, and $\beta$ commutes with augmentations and differentials. Therefore the target map is induced by ${\beta }_{F^{\mathrm{\prime }}(I)}G({\alpha }_I)$ and is the stated derived-composite map. The bicomplex map preserves the filtration, so its cohomology map preserves the image filtration. Combine this construction with the input map in F1; naturality of $\alpha ,\beta$ shows the order of combination agrees.

Identity transformations give identity $E_2$ and target maps. For composites, either lift the composite or compose lifts: they extend the same complex map and F2 gives the same $E_2$ maps; the commuting augmentation squares give the same target map. These facts prove naturality, with exactly the stated DC/supplied-data qualification. Zero transformations and zero objects yield zero maps throughout.