This is an exercise question in Dold Algebraic Topology, Chpt VI, 2.12. The point is to demonstrate that additive functors does not preserve infinite direct sums.
$\textbf{Q:}$ What is the example showing that $Hom_{Ab}(A,-)$ does not preserving infinite direct sums? I tried $A=Z_4, Z_2$ with $-=\oplus Z_4, \oplus Z_2$. However, it seems I cannot get something uncountable out by taking a countable finite direct sums. Any hint will be helpful.