For an abelian group $G$ we denote by $G^*$ the $\mathbb{Z}$-module $\text{Hom}_\mathbb{Z}(G,\mathbb{Q}/\mathbb{Z})$ -- group of all $\mathbb{Z}$-module homomorphisms from $G$ to $\mathbb{Q}/\mathbb{Z}$ (the quotient group).
Now, let $A,B$ and $C$ be abelian groups. Let $0\to B\stackrel{\mu}{\to} A\stackrel{\epsilon}{\to} C\to 0$ be a sequence of homomorphisms.
Suppose $0\to C^*\stackrel{\epsilon^*}{\to} A^*\stackrel{\mu^*}{\to} B^*\to0$ is an exact sequence. Is the sequence $0\to B\stackrel{\mu}{\to} A\stackrel{\epsilon}{\to} C\to 0$ also exact?
I've heard it is, but I have no idea why. I will apreciate any hints and advices how to prove it. (I suppose that injectivity of the $\mathbb{Z}$-module $\mathbb{Q}/\mathbb{Z}$ (it is a divisible group) may be important.)