Let $G$ be an affine algebraic group over a field $k$, of coordinate ring $O(G)$. This is the same as a functor $F$ from the category of $k$-algebras to the category of groups such that for any $k$-algebra $R$, the group $F(R)$ is in (natural) bijection with the set $Hom_{k\text{-alg.}}(O(G), R)$.
Is it true that for any $k$-algebras $R,S$ we have a group isomorphism $$G(R \times S) \cong G(R) \times G(S)$$ ?
I know that we have a bjection $Hom_{k\text{-alg.}}(O(G), R \times S) \simeq Hom_{k\text{-alg.}}(O(G), R) \times Hom_{k\text{-alg.}}(O(G), S)$ but I don't see why it should preserve the group operations.