Eilenberg steenrod uniqueness on finite dimensional cw pairs

F1 constructs the normalized comparison on every finite CW pair and makes it natural for inclusions of finite subcomplex pairs. By F2, take the colimit of these maps over all finite $K\subset X$ to define an isomorphism on a finite-dimensional pair $(X,A)$. Its inverse is the colimit of the inverse comparisons.

Every finite $K$ is compact, so a continuous map $f:(X,A)\to (Y,B)$ carries $K$ into a finite subcomplex $M\subset Y$ by compact-cell support as used in F2. The restricted map $(K,K\cap A)\to (M,M\cap B)$ is a map of finite pairs. Naturality there, followed by the two colimit maps, gives naturality on the class represented in $K$. Every class has such a representative, proving naturality for all continuous maps.

The boundary of a class supported on $K$ is supported on $K\cap A$, and F1 makes the boundary square commute on that finite pair. Therefore it commutes on the colimit. Any other normalized natural morphism agrees on every finite pair by F1, and hence on every class by F2. At a point its component remains $u$. The one-stage five-lemma principle F3 also propagates invertibility of this already constructed morphism through each finite skeletal stage; it is not needed to invent the comparison maps.