text: Q:
  
  a counterexample for the morphism of sheaves
  
  Suppose $\textbf{F}$ and $\textbf{G}$ are two presheaves over a topological space $X$,and $\mu:\textbf{F}\longrightarrow \textbf{G}$ is a morphism of presheaves which is surjective.We have a naturally induced morphism of the associated sheaves $\mu^+:\textbf{F}^+\longrightarrow \textbf{G} ^+$.I feel that $\mu^+$ may not be surjective.Is it right?If so, can you give me an example for this?Thank you very much.
  
  A:
  
  Well, you have the category of presheaves of abelian groups.
  Whenever you have a category you can define "epimorphism" by a universal property (see any textbook or wiki). But one can show that this really is equivalent to surjectivity on the sections.
  
  