Right, he goes on to say "Exactness at $\mathcal{G}$ follows from the next lemma: let $\psi : \mathcal{F} \to \mathcal{K}$ be a sheaf injection with $\psi_x : \mathcal{F}_x \cong \mathcal{K}_x$ for all $x \in X$. Then $\mathcal{F} = \mathcal{K}$" and uses the sheaf axioms to prove this
It's very confusing lol. The next line he says "since taking stalks and taking kernels commute, the above inclusion shows that $\mathcal{F}_x \cong \ker\{\mathcal{G} \to \mathcal{H}\}_x$ for all $x$."
In the notes my prof writes "The injectivity of $\mathcal{F}(U) \to \mathcal{G}(U)$ follows from diagram. One shows analogously that the composition $\mathcal{F} \to \mathcal{H}$ is zero, so that $\mathcal{F} \subset \ker\{\mathcal{G} \to \mathcal{H}\}$."
So we're assuming $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ is exact and we want to show that $0 \to \mathcal{F}(U) \to \mathcal{G}(U) \to \mathcal{H}(U)$ is exact
in the proof my prof shows that $\mathcal{F}(U) \to \mathcal{G}(U)$ is injective (this is easy from some diagram) and then goes on to start proving that $\mathcal{F} \to \mathcal{H}$ is the zero map and thus that $\mathcal{F}$ is contained in its kernel, and hence that $\mathcal{F}_x \cong \mathcal{H}_x$ for all $x$ (the stalks)
Here goes: a sequence of sheaves $0 \to \mathcal{F} \to \mathcal{G}, \to \mathcal{H} \to 0$ is exact if and only if $0 \to \mathcal{F}(U) \to \mathcal{G}(U) \to \mathcal{H}(U)$ is exact and some other condition whose proof I understand
A good example is Faroese, where the official orthography was only actually written down in the 1800s some time and the orthography was completely based on old Norse etymology, making the language more or less entirely unphonetic
