Additivity and compact cell support control the infinite cw colimit

Use the telescope from F2. Subdivide its height intervals at half-integers. Let $B_0=X^0\times [0,1]$, and for $i\ge 1$ let $B_i=(X^{i-1}\times [i-1\mathrm{/}2,i])\cup (X^i\times [i,i+1])$. Even $B_i$ form a disjoint-union subcomplex $U$, and odd $B_i$ form a disjoint-union subcomplex $V$; they cover the telescope. Their intersection is the disjoint union $C_i=X^i\times [i+1\mathrm{/}2,i+1]$. Each $B_i$ retracts onto $X^i$ at height $i$, and each $C_i$ onto $X^i$, with all retractions preserving the corresponding pieces of $T_A$.

Apply relative Mayer–Vietoris F1 and arbitrary additivity. After the retractions, the overlap map has from the $i$th summand the identity into stage $i$ and the inclusion-induced map into stage $i+1$, with opposite signs. Multiplying each odd-indexed overlap summand by $-1$, and reordering the target even/odd direct sums by stage, identifies it with $1-s$ on ${⨁}_i{h}_n(X^i,A^i)$. These changes of signs leave the outgoing sum map equal to the canonical stage-to-telescope map.

F3 says $1-s$ is injective also in degree $n-1$. Exactness therefore makes the map from the stage direct sum onto telescope homology surjective, with kernel $\mathrm{im}(1-s)$. Its cokernel is the sequential colimit by F3. F2 identifies telescope homology with $h_n(X,A)$ by the actual projection. Its composite on every stage is the canonical inclusion, so the isomorphism obtained is the claimed canonical map.

Each skeletal pair is finite-dimensional, so F4 identifies its group with the colimit of the finite subcomplex pairs it contains. Every finite CW subcomplex of X lies in some skeleton, as its finitely many cells have bounded dimensions. Thus the iterated colimit is precisely the colimit over all finite subcomplex pairs, proving that assertion.

Let $f:(X,A)\to (Y,B)$ be any continuous map of CW pairs, and write $j_X$ and $j_Y$ for the canonical finite-subcomplex colimit isomorphisms of step 4.1. Define the map between these colimits by $j_Y^{-1}\circ f_{\ast }\circ j_X$. This is well defined and functorial because $j_X,j_Y$ are canonical isomorphisms and ordinary homology is functorial. On a representative class from a finite subcomplex pair of $X$, it is exactly the finite-support representative of its image under $f_{\ast }$ in $Y$ furnished by step 4.1. Thus $j_Y$ composed with this map equals $f_{\ast }{j}_X$, proving the stated naturality for arbitrary maps without requiring a map to preserve skeleta or a finite image subcomplex. Empty $X$ and all zero groups give zero colimits, and all degrees, including negative ones, are covered by the same exact sequences.