I'm working through Vakil's notes about algebraic geometry right now, still in the first chapters, and one of the main results in section 2.5 is, that "Exactness of sequences can be checked on stalks". Then one shows that taking the stalk at a point $p$ is an exact functor. But what does that sentence mean? Sure, if have a exact sequence of sheaves I will get a exact sequence of stalks. But what about the other direction? Sure, taking stalks and taking images and kernels and whatnot commute, but how do we get that for example if $ker(\phi)_p=im(\psi)_p$ holds for all $p$ that also $ker(\phi)=im(\psi)$ is true in the category of sheaves? I'm hesitant because he writes in his notes before that an isomorphism on all stalks doesn't imply an isomorphism of the sheaves.
Maybe it's a naive question but I'm still very new to the subject. A somewhat related question: What about the 'taking sections over $U$'- functor? Since this is left-exact, can i somehow also check left-exact sequences on sections?
Glad for any advice!