I have in my lecture notes written down that if $X$ is an affine variety and $M$ is a $k[X]$ module then the presheaf $\mathscr F$ defined as $\mathscr F(U) = M\otimes_{k[X]}\mathscr O_X(U)$ is equal to the sheaf $M^\sim$. I can see that they agree on the basic open sets so if $\mathscr F$ is a sheaf then we are done, but I can't seem to show that $\mathscr F$ is a sheaf. How would I go about this argument?
Edit: $M^\sim$ is defined as $(\mathscr F|_{\mathcal B})^+$ where $\mathcal B$ is the set of basic open sets.