Need a description of the map $i:\mathbb{C}P^2\vee\mathbb{C}P^2\rightarrow \mathbb{C}P^2\times \mathbb{C}P^2$ on $2$nd Cohomology level

Question

Let $i:\mathbb{C}P^2\vee\mathbb{C}P^2\rightarrow \mathbb{C}P^2\times \mathbb{C}P^2$ be the inclusion map. I want to know what is $i^*$ on $2$nd Cohomology level. Seems it should be an isomorphism between $H^2(\mathbb{C}P^2\vee\mathbb{C}P^2) $ and $H^2(\mathbb{C}P^2\times\mathbb{C}P^2) $. But can't find a rigorous proof. Any help is appreciated.