I'm reading Topological methods for C*-algebras. III. Axiomatic homology.
A cofibre theory is a sequence of (covariant) functors from some subcategory of $C^*$-algebras to abelian groups which satisfies severel axioms, and some of them are:
Cofibre axiom. Let $p:A\to B$ be a cofibration and let $Cp=\{(\xi,a)\in CA\oplus B:\xi(0)=p(a)\}$. Then the sequence $h_n(Cp)\to h_n(A)\to h_n(B)$ is exact for each $n$.
Suspension axiom. $h_n(A)$ is isomorphic to $h_{n-1}(SA)$ naturally.
Homotopy axiom. Let $f_0,f_1:A\to B$ be homotopic morphisms, then $h_n(f_0)=h_n(f_1)$.
$IA,CA,SA$ stands for $C([0,1])\otimes A$ , $C([0,1))\otimes A$ , $C((0,1))\otimes A$ respectively.
A cofibre sequence is a sequence $A_n$ where the morphisms $A_n\to A_{n-1}$ are cofibrations. $p:A\to B$ is called a cofibration when the following holds:
Let $h:D\to IB$ be a homotopy from $h_0$ to $h_1$. Assume there is $f:D\to A$ such that $pf=h_0$, then there is homogopy $H:D\to IA$ such that $pH=h$ with $H_0=f$.
Above are the definitions. And proposition 3.8 says $h_n(A_{k+1})\to h_n(A_k)\to h_n(A_{k-1})$ is exact whenever $h_*$ is a cofibre theory and $A_*$ is a cofibre sequence.
The proof says it is immediate from cofibre axiom but I don't see why. Denote by $C_k$ the mapping cone (cofibre) of $f_k:A_k\to A_{k-1}$, that is, $C_k=\{(\xi,a)\in CA_{k-1}\oplus A_{k}: \xi(0)=f_k(a)\}$. Cofibre axiom says that $h(C_k)\to h(A_k)\to h(A_{k-1})$ is always exact. But how do I show the image of $h(C_k)\to h(A_k)$ and that of $h(A_{k+1})\to h(A_k)$ are the same?