Let $\phi : F \rightarrow G$ be a map of sheaves. Let $\phi _p : F_p \rightarrow G_p$ be the induced maps of stalks at a point p $\in$ X (where F and G are sheaves over some topological space X)
Show that if $\phi_p$ is surjective for every p in X, then $\phi$ is surjective.
i.e show that if Im($\phi_p$)$= G_p$ for every p then Im($\phi$)$ = G$.
This is in reference to Hartshorne Chapter 2, exercise 1.2b.
I have shown that (Im$\phi$)$_p = $ Im($\phi_p$) and so if the map on stalks is surjective then we have that (Im$\phi$)$_p =G_p$
But I'm not sure how to proceed from there. Most of the proofs I have found just assert that this implies that Im($\phi$)$ = G$ (which I don't see how it follows, because from what I understand, equality on stalks does not imply equality of sheaves). Some others involve a manipulation of direct limits and co-kernels but I don't quite understand how the proof for this works(I am not very fluent with category theory).
Is there a way to prove this without (too much) category theory? If not, I'd appreciate any help with the category theoretic proof.
Edit: here Im $\phi$ is the sheafification of the image presheaf