kernel of certain sheaf morphisms

Question

Let $C$ be a site, $Set$ be the category of sheaf of sets on $C$, $Ab$ be the category of sheaves of abelian groups on $C$. There is a left adjoint functor $\mathbb{Z}[-]$ to the forgetful functor $Ab \to Set$. More explicitly, it takes $\mathcal{F} \in Set$ to the sheaf associated to the presheaf $U \to \mathbb{Z}[\mathcal{F}(U)]$. Given an $\mathcal{F} \in Ab$, there is a surjection (in $Ab$) $\mathbb{Z}[\mathcal{F}] \to \mathcal{F}$. How's the kernel look like?