Where is sheafification in the definition of exact sequence of sheaves? I am reading Andreas Gathmann's notes on Algebraic geometry,http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf
Def 7.1.14(iv)says the following 

As usual, a sequence of sheaves and morphisms
  $$ \cdots \rightarrow F_{i−1} \rightarrow F_i \rightarrow F_{i+1} \rightarrow \cdots $$
  is called exact if ker$( F_i \rightarrow F_{i+1}) = $im$( F_{i−1}\rightarrow F_i ) $ for all $i$.

However，even every arrow is a morphism of sheaves, the image of a sheaf will not necessarily be a sheaf, so what does this definition really mean? I have two guesses.
One is that if one of the image is not a sheaf, then the sequnce is never exact. The other is that the notation im$( F_{i−1}\rightarrow F_i ) $ stands for the sheafification of the presheaf image.
Which one is true? (or both are wrong?)
 A: Sheaves of abelian groups form an abelian category, and the definition given is the usual definition of the exactness of a sequence in an abelian category. Image here means the categorical image in an abelian category, which is equivalently either the kernel of the cokernel or the cokernel of the kernel; either way, to compute it you have to compute a cokernel, and that cokernel can be computed as a presheaf cokernel and then sheafified. 
A: In any Abelian category, the image of a morphism is the kernel of the cokernel.
We can use the relationship bewteen sheaves and presheaves to compute the image, however: if $i,a$ are the inclusion and sheafification functors, then if $f$ is any morphism:
$$ \begin{align}\mathop{\mathrm{im}} f
&= \ker \mathop{\mathrm{coker}} f 
\\&\cong \ker ai(\mathop{\mathrm{coker}} f) 
\\&\cong a(\ker \mathop{\mathrm{coker}}if)
\\&\cong a(\mathop{\mathrm{im}} (if))
\end{align}$$
using the fact that $ai \cong 1$, $a$ is left exact, and $i$ is right exact. Thus, the image of a sheaf morphism is indeed the sheafification of its image when viewed as a presheaf morphism.
Note, for example, $\mathop{\mathrm{im}} (f)$ means the image as a sheaf morphism, and $\mathop{\mathrm{im}} (if)$ means the image as a presheaf morphism. Allow me to emphasize that the result of this calculation literally is the image of a morphism in the category of sheaves of abelian groups.
