Let $S$ be an abelian semigroup. The Grothendieck group of $S$ is $K(S)=S\times S/\mathord{\sim}$, where $\sim$ is the equivalence relation: $(s,t)\sim(u,v)$ if there exists $r\in S$ such that $s+v+r=t+u+r$. This is indeed an abelian group with zero element $(s,s)$ (any $s\in S$), inverse $-(s,t)=(t,s)$ and addition given by $(s,t)+(u,v)=(s+u,t+v)$. It is common to use the suggestive notation $t-s$ for $(t,s)$.

The Grothendieck group construction is a functor from the category of abelian semigroups to the category of abelian groups. A morphism $f\colon S\to T$ induces a morphism $K(f)\colon K(S)\to K(T)$ which sends an element $(s^{+},s^{-})\in K(S)$ to $(f(s^{+}),f(s^{-}))\in K(T)$.

Let $C$ be a (essentially small) symmetric monoidal category. Its Grothendieck group is $K([C])$, i.e. the Grothendieck group of the isomorphism classes of objects of $C$.