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](http://link.springer.com/article/10.1007%2FBF01403148) 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 $\Delta\_\*$?



group completion theorem of homology as Hopf algebras
1. List item $H\_\*(M)[\pi \_0(M)^{-1}]$ inherits its coalgebra structure from
$H\_\*(M)$. i.e., there is a unique coalgebra structure so that the localization
becomes a map of coalgebras. Concretely we "extend" the diagonal by declaring 
elements of $\pi \_0(M)^{-1}$ to be group-like.
2. List item Since the map $M\rightarrow \Omega BM$ is a map of spaces, it induces
a map of coalgebras. So you get an isomorphism of coalgebras.


