Given a family of ring epimorphisms $$ \varphi_i\colon R_i \to S_i $$
is the unique ring morphism
$$ \varphi \colon R \to S $$ with $R=\prod_i R_i$, $S=\prod_i S_i$ and $\pi^S_i \circ \varphi = \varphi_i \circ \pi^R_i $ an epimorphism? The $\pi$ denote the projections of the products.