There's something bugging me on the theory of sheafification.
I proved that, given $P$ a presheaf on a topological space $X$ , there exists a sheaf $P^*$ and a presheaf homomorphism $f:P\to P^*$ such that for every sheaf $F$ and homomorphism $\varphi:P\to F$ there's a unique $\tilde{\varphi}:P^*\to F$ such that $\tilde{\varphi}\circ f=\varphi$.
But the theorem states also that the following are equivalent:
$\tilde\varphi$ isomorphism
$\forall x\in X$ the stalks are isomorphic $\varphi_x:P_x\cong F_x$
$\varphi$ is locally injective and surjective
I proved also that $P^*_x\cong P_x$ so it's easy to see that the first two are equivalent. But I have no idea why the last one should be equivalent to the other.. I don't know whether it's wrong or I have the wrong definition for "locally inje/surjective" or simply I'm missing totally the point..
P.s. On a total different side.. how do I draw diagrams here? In Latex I use tikz, but I don't know if it works even here.