Tensor product of the structure sheaf with the function field

Question

Consider an irreducible scheme $X$ with function field $K(X)$. Then define the presheaf

$$U\mapsto \mathscr O_X(U)\otimes_{\mathscr O_X(U)} K(X)$$

for every open set $U\subset X$.

Is this presheaf actually a sheaf? I know that in general in order to construct the tensor product of two $\mathscr O_X$-modules we need a sheafification process. I'm asking if in this case we can avoid the sheafification.

Answer 1

Hint: This presheaf assignes to any nonempty $U$ the group $K(X)$. Furthermore, the restriction maps should be the identity on $K(X)$. This should help in showing this is already a sheaf.