homomorphism between cohomology induced by the multiplication of an H-space Define the product on $\mathbb{C}P^\infty$ in the following way:
\begin{eqnarray*}
\phi:\mathbb{C}P\overset{\Delta}\longrightarrow(\mathbb{C}P^\infty)^k\overset{\mu}\longrightarrow \mathbb{C}P^\infty,\\
y\mapsto (y,y,\cdots,y)\mapsto y^k.
\end{eqnarray*}
Here $\Delta$ is the $k$-diagonal map and $\mu$ the multiplication given on Hatcher's book algebraic topology page 282.
Then
\begin{eqnarray*}
\phi^*: H^*(\mathbb{C}P^\infty;\mathbb{Z})&\to& H^*(\mathbb{C}P^\infty;\mathbb{Z}),\\
\mathbb{Z}[\alpha]&\to& \mathbb{Z}[\alpha].
\end{eqnarray*}
What is $\phi^*\alpha$? How to compute?
And
\begin{eqnarray*}
\phi_*: H_*(\mathbb{C}P^\infty;\mathbb{Z})&\to& H_*(\mathbb{C}P^\infty;\mathbb{Z}),\\
\Gamma_\mathbb{Z}[\alpha]&\to& \Gamma_\mathbb{Z}[\alpha].
\end{eqnarray*}
What is $\phi_*\alpha$? How to compute?
 A: It suffices to know the comultiplication induced by $\mu$ on cohomology. For degree reasons the unique possibility is
$$\Delta(\alpha) = \alpha \otimes 1 + 1 \otimes \alpha$$
and from here you are in the realm of a purely Hopf-algebraic calculation: you want to apply the "$n^{th}$ comultiplication" $\Delta_n : H \to H^{\otimes n}$ (where $H$ is the Hopf algebra in question, here the cohomology) and then apply the "$n^{th}$ multiplication" $m_n : H^{\otimes n} \to H$. By $m_n$ I mean the operation
$$\alpha_1 \otimes \dots \otimes \alpha_n \mapsto \alpha_1 \dots \alpha_n$$
and by $\Delta_n$ I mean the dual thing. The result is called the $n^{th}$ Adams operation $\psi_n$ and can be computed in this case as follows. We have
$$\Delta_n(\alpha) = \sum_{k=0}^n 1 \otimes \dots \otimes \alpha \otimes \dots \otimes 1$$
where in the $k^{th}$ term $\alpha$ appears in the $k^{th}$ position, and hence
$$\psi_n(\alpha) = m_n \Delta_n(\alpha) = \sum_{k=0}^{n-1} \alpha = n \alpha.$$
The Adams operations are all ring homomorphisms, so this uniquely determines $\psi_n$ on the entire cohomology ring. The computation for homology is dual to this one. 
