Let $R$ and $S$ be unital rings. For every unital ring $T$, let $K_0(T)$ denote the group completion of the commutative monoid $V(T)$ of isomorphism classes of finitely generated projective left $T$-modules under direct sum. Then the assignment
The naturality is with respect to pairs of unital ring homomorphisms $\alpha:R \to R'$ and $\beta:S \to S'$ through the induced homomorphism $\alpha \times \beta:R\times S\to R'\times S'$.