In an arbitrary category, we have that even if $X$ and $Y$ have a product $X \times Y$, the natural projections needn't be epimorphisms.
Are there (preferably simple!) conditions we can place on the category such that all the natural projections are, in fact, epimorphisms?
Without assuming anything about the category, is there a (preferably interesting!) weaker property that the natural projections always possess?