As I promised I would do some time ago (in a comment on Pierre-Yves Gaillard's answer), and with the hope this is useful, I have written an <a href="http://nlab.mathforge.org/nlab/show/coproduct-preserving+representable">article</a> in the nLab which ties together some of the results in the answers already given into a single narrative. Here and there I provide a few extra glosses and some background information.