This is a continuation of a previous question.
Is it possible to find a map $S : \mathrm{Set} \times \mathrm{Set} \to \mathrm{Set}$ such that $S(X,Y)$ is a coproduct of $X$ and $Y$ (thus it is equipped with universal morphisms $X \rightarrow S(X,Y) \leftarrow Y$; also $S$ will be a functor) and that $S$ is strictly commutative in the sense that the canonical bijection $S(X,Y) \cong S(Y,X)$ is the identity?