I want to prove the following:
Let $\varphi: \cal{F} \longrightarrow\cal{G}$ a morphism of presheaves such that $\varphi (U): \cal{F}(U) \longrightarrow\cal{G}(U)$ is injective for each $U$. Then the induced map of associate sheaves $\varphi ^+: \cal{F}^+ \longrightarrow\cal{G}^+$ is injective.
I know that $(Ker \phi )_p=Ker(\phi_p)$ if $\phi$ is a morphism of sheaves and that sheafification preserves stalks. If $\varphi_p$ is injective $\forall p$ then $\varphi^+_p$ is injective $\forall p$ and then $ker \varphi^+=0$. But how can I prove that $\varphi_p$ is injective $\forall p$, being $\varphi$ a morphism of presheaves, not sheaves?