What can we conclude about the natural projection maps? 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.
Two questions:


*

*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?

You might like this theorem:
If $\pi _i:P {\to} A_i$ for $i\in I$ is a product and if $i_0 \in I$ is such that, for each $i \in I$, $hom(A_{i0} ,A_i)$ is not empty 
then $\pi _{i0}$ is a retraction.
In general the $\pi _i$ form an extremal mono-source
See: 
Abstract and concrete categories: the joy of cats. Propositions 10.21 and 10.28
