Let $f\colon A\to B$ be a group homomorphism between finite abelian groups.
For abelian group $G$, let $G^\wedge=\operatorname{Hom}_\mathbb{Z}(G,\mathbb{Q}/\mathbb{Z})$ be its Pontryagin dual.
Since $A,B$ are finite abelian groups, we have $A^\wedge=A, B^\wedge=B$.
Now, my question is :
How can we describe $f^\wedge\colon B\to A$ in terms of $f$?
Added: Can we relate $\operatorname{Ker} f^\wedge$, $\operatorname{Coker} f^\wedge$ with $\operatorname{Ker} f $, $\operatorname{Coker} f$?