In the first chapter of Ravi Vakil's Algebraic Geometry notes, he suggests a construction of the groupification $H(S)$ of an abelian semigroup $S$ by considering $S\times S/\sim$ where $(a,b)\sim (c,d)$ iff there is $e\in S$ such that $a+d+e=b+c+e$. 

I can verify that this is a group, but I can't figure out what the universal map $S\to H(S)$ should be so that all morphisms $S\to G$ for $G$ a group factor uniquely though $H(S)$. For the Grothendieck group of a commutative monoid, the same construction would have the universal map $s\mapsto (s,0)$, but I don't know how to mimic this if $S$ has no identity.