Naturality and edge maps of the AHSS

Since $f$ is cellular, it maps $X^p$ into $Y^p$ for every $p$, so on each pair $(X^p,X^{p-1})$ it induces a map of pairs and hence a map $h^n(Y^p,Y^{p-1})\to h^n(X^p,X^{p-1})$ and, in the absolute groups, $h^n(Y^p)\to h^n(X^p)$. By naturality of the pair long exact sequences and of the suspension isomorphisms these maps commute with $i,j,k$ and define a morphism of the cohomological skeletal exact couples; the homological case is the same with covariant arrows.

Identity cellular maps induce identity exact-couple morphisms, and composites of cellular maps induce the composites of the corresponding exact-couple morphisms, by functoriality on every skeletal pair.

A morphism $\phi$ of reduced generalized cohomology theories commuting with suspension and connecting maps assigns to each pair long exact sequence a commuting ladder, hence defines a morphism of cohomological skeletal exact couples; the same argument for a morphism of reduced generalized homology theories gives a morphism of homological skeletal exact couples.

By [F2] the morphisms of step 1.1 and step 1.3 induce morphisms of the derived couples and of every page, commuting with differentials and transitions; composition and identities are respected because the exact-couple construction is functorial.

The induced morphisms carry the stable numerator and denominator of the exact couple into the corresponding stable subobjects, so under the edge identifications of [F4] they induce the maps on the filtration stages and filtration quotients; hence the edge maps are preserved.

Steps 1.2, 2.1 and 3.1 show that the induced page maps respect identities and composition, commute with all structure, and preserve the edge maps, which proves the theorem in both the cohomological and homological cases.