Let $\Omega^{n+1}S^{n+1}$ be the base-pointed $(n+1)$-iterated loop space on the $(n+1)$-sphere. In the paper The homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, Lecture notes in mathematics Vol. 533, page 226, Theorem 3.2, it is proved that as a $AR_n\Lambda_n$-Hopf algebra, $$ H_*(\Omega^{n+1}S^{n+1};\mathbb{Z}_p)\cong GW_n H_*(S^0;\mathbb{Z}_p) $$ for primes $p\geq 2$.

Question: In page 223, it is said that "the coproduct are determined by the diagonal Cartan formulas (page 213, (4))". I do not understand how to write the coproduct of $H_*(\Omega^{n+1}S^{n+1};\mathbb{Z}_p)$ explicitly?

In the paper On the homology of configuration spaces, Section 4, as an associative and commutative $\mathbb{Z}_p$-algebra, a generator set of $H_*(\Omega^{n+1}S^{n+1};\mathbb{Z}_p)$ is given. I want to know how to obtain the coproduct formulas of $H_*(\Omega^{n+1}S^{n+1};\mathbb{Z}_p)$ as a coalgebra.