The complex K-theory spectrum $KU$ is equipped with a $C_2$-action coming from conjugation of vector bundles. On homotopy, we have that $\pi_*KU = \mathbb{Z}[u^{\pm 1}]$, and the action is recorded by $u \mapsto -u$.
The smash product $KU \wedge KU$ also has a $C_2$-action: just take the previous action on one of the factors in the smash product. Adams, Harris, and Switzer determine the cooperation algebra $\pi_*(KU \wedge KU) = KU_* \otimes KU_0KU$ as a subring of $\mathbb{Q}[u^{\pm 1}, v^{\pm 1}]$.
My question is: how can one see the action on the level of homotopy in the second instance? On the level of generators it should still send $u \mapsto -u$, but on the whole of $KU_0KU$ I do not know how to think about the fixed points of the action.