Assume we are in the category of sheaves of $\mathcal{O}_X$-modules. Suppose two presheaves maps to the same sheaf under the sheafification functor. Does it imply that two presheaves were same?
I am trying to prove that it should be true. If not then two presheaves have different global sections but global sections of presheaf is contained in global sections of its associated sheaf, which are equal. I am unable to progress further. Any help would be great.