Edge homomorphisms are natural

At the vertical homological axis, outgoing maps are zero. The commutative differential squares in [F2] carry each incoming image into its counterpart, so the induced map on its cokernel commutes with the transition quotient. Compatibility with α equates that map with the next page morphism. Compose these squares up to a common finite stabilization page to obtain a square $E_{0,n}^s\to E_{0,n}^{\mathrm{\infty }}$ for source and target.

At the horizontal axis, incoming maps are zero. The same differential squares restrict to outgoing kernels, and α compatibility gives commuting squares for all transition inclusions. Their composite yields the square for $E_{n,0}^{\mathrm{\infty }}\hookrightarrow E_{n,0}^s$.

A compatible filtered map on H commutes with $F_0{H}_n\hookrightarrow H_n$ and $H_n\twoheadrightarrow H_n\mathrm{/}{F}_{n-1}{H}_n$, and compatibility identifies their graded maps with the stable maps. Attach these squares to steps 1.1–1.2 to obtain naturality of both edges. In cohomological indexing the kernel and quotient axes are exchanged, giving the two analogous squares. For filtered chain maps, [F3] and [F4] provide exactly the page and abutment compatibilities just used.