Let $\pi:X\to Y$ be a continuous map, $\pi(p)=q$. Let $\mathcal{G}$ be a sheaf on $Y$. I want to show that there is a natural isomorphism $\mathcal{G}_q \to (\pi^{-1}\mathcal{G})_p$ by using the adjunction of inverse image and pushforward.
To start, we notice that a stalk is a colimit, which commutes with left adjoints (inverse image here). Hence,$$(\pi^{-1}\mathcal{G})_p=\lim_{p\in U}\pi^{-1}\mathcal{G}(U)=\pi^{-1}\lim_{p\in U}\mathcal{G}(U)=\pi^{-1}(\mathcal{G_p}).$$
I don't know how to make sense of $\pi^{-1}(\mathcal{G_p})$ such that it is isomorphic to $\mathcal{G}_q$. Please feel free to show me other ways (by using adjoints) if my approach doesn't make much sense to you.