A comparison isomorphism propagates over one skeleton stage

Given: The objects and hypotheses in the statement above.

For ordinary theories $h,k$ as in def-unreduced-homology-theory-on-cw-pairs, a morphism $\eta :h\to k$ consists of homomorphisms ${\eta }_n(X,A):h_n(X,A)\to k_n(X,A)$, natural for all maps of CW pairs and all $n\in \mathbb{Z}$, satisfying ${\mathrm{\partial }}^k{\eta }_n={\eta }_{n-1}{\mathrm{\partial }}^h$. For a specified homomorphism $u:h_0(\ast )\to k_0(\ast )$, the morphism is coefficient-normalized by $u$ if ${\eta }_0(\ast )=u$. A comparison equivalence has every component invertible and is normalized by a specified coefficient isomorphism. Neither the existence nor uniqueness of such an extension is part of this definition. (Coefficient normalized morphism of ordinary homology theories)

For a CW pair $(X,A)$ and an ordinary theory $h$ with coefficient $G$, put $F_{-1}=A$ and $F_r=A\cup X^r$ for $r\ge 0$. Set $C_r^h(X,A)=h_r(F_r,F_{r-1})(r\ge 0),C_r^h=0(r<0)\mathrm{.}$ Each $C_r^h$ is the direct sum of copies of $G$ indexed by the relative $r$-cells. Define $d_0=0$ and for $r\ge 1$ let $d_r$ be the triple boundary to $h_{r-1}(F_{r-1},A)$ followed by its map to $h_{r-1}(F_{r-1},F_{r-2})$. Then $d_{r-1}{d}_r=0$, naturally for cellular maps of CW pairs. (Any ordinary homology theory has a cellular chain complex on a cw pair)

Let $\cdots \to A_{n-1}\to A_n\to A_{n+1}\to A_{n+2}\to A_{n+3}\to \cdots$ and $\cdots \to B_{n-1}\to B_n\to B_{n+1}\to B_{n+2}\to B_{n+3}\to \cdots$ be long exact sequences in an abelian category, together with a morphism of these sequences. If the four comparison maps at $A_{n-1},A_n,A_{n+2},A_{n+3}$ are isomorphisms, then the comparison map $A_{n+1}\to B_{n+1}$ is an isomorphism. (Five lemma for a morphism of long exact sequences)