I am self-studying little bit of algebraic geometry and I got stuck in a proof I cannot understand. First of all, we define a sequence of sheaves $\mathscr F \stackrel{\theta}{\to} \mathscr{G}\stackrel{\phi}{\to}\mathscr{G}$ to be exact if the corresponding sequence on stalks is exact. According to a Proposition, a sequence of sheaves is exact if and only if Im$\theta$= Ker$\phi$ as subsheaves of $\mathscr G$.
First of all, the image presheaf is a sub-presheaf of $\mathscr G$ but why is Im$\phi$ a subsheaf of $\mathscr G$? And secondly, how can I prove the proposition above? Thank you.