This is part of exercise 6.N in Wegge-Olsen's book '$K$-theory and $C^*$-algebras'. In the following, $A$ and $B$ are $C^*$-algebras, $\alpha:A\rightarrow B$ is a surjective $C^*$ morphism with kernel $J$,
$CB$ denotes the cone of $B$, i.e. $CB=\{f\in C([0,1],B):f(0)=0\}$,
$SB$ denotes the suspension of $B$, i.e. $SB=\{f\in C([0,1],B):f(0)=f(1)=0\}$,
$C_{\alpha}$ denotes the mapping cone of $\alpha$, i.e. $C_\alpha=\{(a,f)\in A\oplus CB:\alpha(a)=f(1)\}$.
There is a morphism $\beta:J\rightarrow C_\alpha,j\mapsto (j,0)$. There is a commutative diagram of exact sequences
The problem is to produce an exact sequence $0\rightarrow CJ\rightarrow C_\beta\rightarrow SCB\rightarrow 0$, and I'm stuck with this. I thought of the map $CJ\rightarrow C_\beta, f\mapsto(f(1),\beta\circ f)$ but I can't finish up.
If we let $\pi$ be the map $C_\alpha\rightarrow CB,(a,f)\mapsto f,$ in the diagram above, I considered the map $C_\beta\rightarrow SCB,(j,f)\mapsto \pi\circ f$. I worked out exactness at $C_\beta$ though I'm not sure if I did it right, and I'm not sure about it being surjective.