Is a homomorphism out of a free abelian group determined by its value at the basis elements?
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Yes, this is what free abelian groups are designed to do. In more details, if $A$ is a free abelian group on the set $S$ then, for all abelian groups $A'$ there is a natural bijection between the set of functions $f:S\to A'$ and the set of group homomorphisms $\psi: A\to A'$.