The definition of sheaf $K^*/\mathcal{O}^*$

Question

The sheaf $K^*/\mathcal{O}^*$ is defined as the sheafification of the cokernel from the inclusion $\mathcal{O}^*\to K^*$. For the pre sheaf $(K^*/\mathcal{O}^*)^{pre}$, $(K^*/\mathcal{O}^*)^{pre}(U):=K^*(U)/\mathcal{O}^*(U)$, then any holomorphic function on $U$ is trivial on $(K^*/\mathcal{O}^*)^{pre}(U)$, but after sheafification a holomorphic function on $U$ $K^*/\mathcal{O}^*(U)$ may not be trivial since otherwise the effective divisors from $H^0(X,K^*/\mathcal{O}^*)$ are all the same, I wonder how to see after sheafification, it makes this difference?