Eilenberg steenrod uniqueness on finite dimensional cw pairs

For every finite subcomplex $K\subset X$, the intersection $K\cap A$ is a finite subcomplex, so F1 gives a comparison $h_n(K,K\cap A)\to k_n(K,K\cap A)$. Naturality for inclusions makes these maps a morphism of directed systems. F2 for $h$ and $k$ identifies their colimits with the respective groups on $(X,A)$, and hence defines the required isomorphism there. This uses only finite chains of cells to represent a class, not a chosen exhaustion of $X$.

Every finite $K$ is compact, so a continuous map $f:(X,A)\to (Y,B)$ carries $K$ into some finite subcomplex $M\subset Y$ by F3. Since $f(K\cap A)\subset M\cap B$, the restriction is a map of finite CW pairs $(K,K\cap A)\to (M,M\cap B)$. F1 makes its comparison square commute. Every class on $(X,A)$ comes from some such $K$ by F2, proving naturality for the original continuous map. A different choice of $M$ gives the same result after inclusion into the finite union of the two choices.

The boundary of a class supported on $K$ is supported on $K\cap A$, and F1 makes the boundary square commute there. The pairs $(K\cap A,\mathrm{\varnothing })$ are cofinal among finite subcomplex pairs of $(A,\mathrm{\varnothing })$, because any finite subcomplex of $A$ can itself serve as $K$. Thus the boundary square commutes after passing to the colimits. Any other normalized boundary-compatible natural morphism agrees with this one on every finite pair by F1, and every class of a finite-dimensional pair comes from a finite pair by F2; hence that morphism agrees everywhere. On a point the construction is $u$.