TITLE: A Question About the Elliott-Natsume-Nest Proof of Bott Periodicity
QUESTION [6 upvotes]: In Wegge-Olsen’s book K-Theory and C$ ^{*} $-Algebras, there is an outline of a proof of Bott Periodicity (the proof is due to George Elliott, Toshikazu Natsume and Ryszard Nest). The first step of the proof outline goes as follows:

Suppose that there are natural transformations

$ \Phi^{0} $ from the $ K_{0} S $-functor to the $ K_{1} $-functor and
$ \Phi^{1} $ from the $ K_{1} S $-functor to the $ K_{0} $-functor,

where $ S $ denotes the suspension functor. In other words, for every C$ ^{*} $-algebra $ \mathscr{A} $ (not necessarily unital), there are abelian-group homomorphisms
  $$
\Phi^{0}_{\mathscr{A}}: {K_{0}}(S(\mathscr{A})) \to {K_{1}}(\mathscr{A})
\quad \text{and} \quad
\Phi^{1}_{\mathscr{A}}: {K_{1}}(S(\mathscr{A})) \to {K_{0}}(\mathscr{A})
$$
  such that for every C$ ^{*} $-algebraic homomorphism $ \alpha: \mathscr{A} \to \mathscr{B} $, the following diagrams commute:
  $$
\require{AMScd}
\begin{CD}
{K_{0}}(S(\mathscr{A}))      @>{{K_{0} S}(\alpha)}>> {K_{0}}(S(\mathscr{B})) \\
@V{\Phi^{0}_{\mathscr{A}}}VV                         @VV{\Phi^{0}_{\mathscr{B}}}V \\
{K_{1}}(\mathscr{A})         @>>{{K_{1}}(\alpha)}>   {K_{1}}(\mathscr{B})
\end{CD}
\quad \quad \quad
\begin{CD}
{K_{1}}(S(\mathscr{A}))      @>{{K_{1} S}(\alpha)}>> {K_{1}}(S(\mathscr{B})) \\
@V{\Phi^{1}_{\mathscr{A}}}VV                         @VV{\Phi^{1}_{\mathscr{B}}}V \\
{K_{0}}(\mathscr{A})         @>>{{K_{0}}(\alpha)}>   {K_{0}}(\mathscr{B})
\end{CD}
$$
  Suppose further that $ \Phi^{0}_{\mathbb{C}} $ and $ \Phi^{1}_{\mathbb{C}} $ are isomorphisms. Then prove that $ \Phi^{0}_{\mathscr{A}} $ and $ \Phi^{1}_{\mathscr{A}} $ are isomorphisms for every C$ ^{*} $-algebra $ \mathscr{A} $.

Wegge-Olsen’s hint is to consider the commutative diagrams above for the homomorphism
\begin{align*}
\alpha: \mathbb{C} & \to     \mathscr{A} \otimes \mathbb{K}; \\
        1          & \mapsto p \otimes e_{11},
\end{align*}
where $ p $ is a fixed choice of a projection in $ \mathscr{A} $ and $ e_{11} $ is a rank-$ 1 $ projection in $ \mathbb{K} \stackrel{\text{def}}{=} {M_{\infty}}(\mathbb{C}) $.

My difficulty:
I do not see how
\begin{align*}
{K_{0}}(\alpha):   & {K_{0}}(\mathbb{C})    \to {K_{0}}(\mathscr{A} \otimes \mathbb{K}), \\
{K_{1}}(\alpha):   & {K_{1}}(\mathbb{C})    \to {K_{1}}(\mathscr{A} \otimes \mathbb{K}), \\
{K_{0} S}(\alpha): & {K_{0}}(S(\mathbb{C})) \to {K_{0}}(S(\mathscr{A} \otimes \mathbb{K}))
\quad \text{and} \\
{K_{1} S}(\alpha): & {K_{1}}(S(\mathbb{C})) \to {K_{1}}(S(\mathscr{A} \otimes \mathbb{K}))
\end{align*}
are isomorphisms. If this hurdle can be overcome, then $ \Phi^{0}_{\mathscr{A}} $ and $ \Phi^{1}_{\mathscr{A}} $ are clearly isomorphisms.
Thank you for any assistance!

REPLY [3 votes]: A vague guess:
Consider something by putting your two Diagramms together (for the map $\alpha:\mathbf{C} \rightarrow A$) so that they fuse to one diagram with three lines. In the first diagram you replace $A$ by $S(A)$ so that double Suspension $SS(A)$ is involved.
The first column is then an isomorphism $K_0(SS(\mathbf{C}) \cong K_0(\mathbf{C})$.
Chase in the Diagramm, to see that the second column is also something (a isomorphism (??!)) for the subgroup generated by $[p]$. Then vary over all $p$.
Maybe something like this.. or a starting Point to think further.