direct product of R-modules not satisfying the universal property of direct sum I found a case in which the (external) direct sum of R-modules do not satisfy the universal property of the direct product of R-modules.
However, I can't think of one in which the direct product of R-modules do not satisfy the universal property of the direct sum of R-modules....
Could anyone suggest any counter example?

Any object satisfying a universal property is unique up to isomorphism.  
A countable direct sum of $\mathbb{Z}$ is not isomorphic to a countable direct product of $\mathbb{Z}$ since they have different cardinalities.  Hence neither satisfies the universal property of the other.  
