Prove of Sheafication of sheaf is a sheaf isomorphism I want to prove in the process of sheafication of presheaf $\mathfrak{F}$, 
If $\mathfrak{F}$,  is a sheaf, then sheafication gives a sheaf isomorphism. 
Can you give a proof of this statement or any reference related with this statement? 
 A: I quote from Hartshorne: 
Definition/ Existence of Sheafification: " Given a presheaf $\mathcal F$, there is a sheaf $\mathcal F ^+$ ,and a morphism of presheafes $\theta: \mathcal F \rightarrow  \mathcal F ^ +$ with the property that for any sheaf $\mathcal G$ and any morphism of sheaf $\phi ; \mathcal F \rightarrow  \mathcal G $(or presheaf), there is a unique morphism $\psi : \mathcal F ^ + \rightarrow G$ such that $\phi= \psi \circ \theta$."   $\mathcal {F}^+$ is called the sheaf associated (or the sheafification) to the presheaf $\mathcal {f}$
By, this property it has very much clear $(\mathcal F ^ +, \theta)$ is unique upto unique isomorphism.  
Now if one take $\mathcal {F}$ itself to be sheaf, it is very obvious that $(\mathcal {F}, Id)$ satisfies the above condition. (i.e., $\mathcal F ^+ = \mathcal F$ and $\theta = Id)$. As the sheafification is unique upto unique isomorphism we conclude $\mathcal F$ is the sheafification of $F$. Identity map is an isomorphism, so we get an isomorphism of the sheaf to its sheafification. 
