I am trying to prove that the inverse image and direct image functors are adjoint.
Let $f : X \rightarrow Y$ be a continuous map. Given sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}$ on $Y$, and a morphism $\varphi : f^{-1}\mathcal{G} \rightarrow \mathcal{F}$, we get a morphism $$\varphi^{\flat} : \mathcal{G} \rightarrow f_* f^{-1} \mathcal{G} \stackrel{f_* \varphi}{\rightarrow} f_* \mathcal{F}.$$ Given a morphism $\psi : \mathcal{G} \rightarrow f_* \mathcal{F},$ we get a morphism $\psi^{\sharp} : f^{-1} \mathcal{G} \rightarrow \mathcal{F}$ which is given on stalks by $$\psi^{\sharp}_x : \mathcal{G}_{f(x)} \rightarrow \mathcal{F}_x, \; \; [U,s] \mapsto [f^{-1}(U), \psi_U(s)], \; s \in \mathcal{G}(U), \; f(x) \in U \subseteq Y \; \mathrm{open}.$$
I have never seen a reference that proves that these operations are really inverse to each other. The most obvious thing to do is to check that $(\varphi^{\flat})^{\sharp}$ and $\varphi$, and $(\psi^{\sharp})^{\flat}$ and $\psi$, are equal on stalks. However, I was only able to verify that $(\psi^{\sharp})^{\flat}$ and $\psi$ are equal on stalks at points $y = f(x)$ for some $x \in X.$ For other points I am stuck.