With regard to Mariano's answer, I believe some clarification is in order. A closely related question was asked by Michael Barr and answered by user Ralph <a href="http://mathoverflow.net/questions/132073/homomorphisms-from-powers-of-z-to-z/132083#132083">here</a>. In brief, the homomorphism named in Martin's question is in fact an isomorphism, provided that $I$ has cardinality less than the first measurable cardinal. 

Shelah and Str&uuml;ngmann (accessible <a href="http://arxiv.org/pdf/math/0009045v1.pdf">here</a>) refer to this result as well, using the same source given by Ralph, just before Definition 2.1: 

> For generalizations to products of larger cardinalities and the resulting definition of slenderness for abelian groups we refer to [EM] or [F1] 

where [EM] is the text by Eklof and Mekler. It seems that Shelah and Str&uuml;ngmann are talking about something slightly different: homomorphisms out of *free complete products* (but using a notation which could unfortunately suggest direct products).