For a topological space $X$, the category of sheaves on $X$ with its values in $Ab$ will form an Abelian Category.
Q1: Is it difficult to prove this?
Next, for the short exact sequence $0 \to F \to G \to H \to 0$, its exactness is usually stated in terms of their stalks at each point $x$ in $X$.
However, somebody told me that this is a "theorem" rather than definition. Namely, once I know that the category of sheaves on $X$ with values in $Ab$ (we call this $Sh_X$) makes an Abelian category, automatically the notion of exact sequence exists.
Hence, it only turns out that the short exact sequence defined via the characteristic of $Sh_X$ being Abelian category is equivalent to the exactness of the given short exact sequence after taking stalk at an arbitrary point $x$ in $X$.
Q2. I cannot see at all what this will mean. Please explain more plainly.
I heartily wish somebody's explanation.