For those who don't have the book to hand: let $E$ be a commutative ring spectrum such that $E_\ast(E)$ is flat as a right module over $E_\ast=E_\ast(*)$, and let $X$ be any spectrum.
Let $\psi_X\colon E_\ast(X)\to E_\ast(E)\otimes_{E_\ast} E_\ast(X)$ be the coaction map, and let $c\colon E_\ast(E)\to E_\ast(E)$ be the conjugation map (which switches the left and right counits).
Then for any elements $a\in E^\ast(E)$, $y \in E^\ast(X)$ and $u\in E_\ast (X)$ with $\psi_X(u) = \sum_i e_i\otimes u_i$, we have $$a\cdot u = \sum_i \langle a,ce_i\rangle u_i$$ and $$\langle ay,u\rangle = \sum_i (-1)^{|y||e_i|}\langle a,e_i\langle y,u_i\rangle \rangle,$$ where $\langle \cdot , \cdot \rangle\colon E^\ast(X)\otimes_{E_\ast} E_\ast(X)\to E_\ast$ is the Kronecker pairing.