An isomorphism theorem for sheaves.

Question

Let $\varphi: \cal{F} \longrightarrow \cal{G}$ a morphism of sheaves. My goal is to prove that $im\varphi \simeq \cal{F} / Ker \varphi$.

My thoughts about this problem:

1) $im \varphi(U) \simeq \cal{F}(U) / Ker \varphi(U)$ (a classic result for abelian groups).

2) We have an isomorphism between sheaves iff we have an isomorphism between their stalks.

Answer 1

Your thought number 1) implies that the presheaf $\mathcal{F}/\ker\varphi$ is isomorphic to the presheaf image of $\varphi$. Hence, the sheafifications of both presheaves are isomorphic, and we're done.