Let $M$ be a topological monoid with product $\mu$. Then $H_*(M)$ is a Hopf algebra with product $\mu_*$ and coproduct $\Delta_*$. The group-completion theorem by McDuff-Segal, 1976 gives that as a Pontrjagin ring, the localization $$H_*(M)[\pi_0M^{-1}]$$ is isomorphic to $$ H_*(\Omega BM)$$ through the map on homology induced by the canonical map $M\to \Omega BM$. It is a ring isomorphism mapping $\mu_*$ to the product of $H_*(\Omega BM)$.

Question:

(1). What is the coproduct structure of the localization $$H_*(M)[\pi_0M^{-1}]$$ induced by $\Delta_*$?

(2). Is the above ring isomorphism $$ H_*(M)[\pi_0M^{-1}]\to H_*(\Omega BM)$$ also an isomorphism of coalgebras preserving the coproduct $\Delta_*$?