So adjoint functors are one thing that has caused me endless pain since trying to learn algebraic geometry. They still haven't really "clicked" like they should have. I've been told that it makes it easier to work with the counit and unit of adjunction, but I tend to find these even harder.
I was trying to do what I assume is a common exercise (exercise II 5.5.3 of Hartshorne): Let $X = \text{Spec }A$ be an affine scheme and show that for any $A$-module $M$ and any sheaf of $\mathcal{O}_{X}$-modules $\mathcal{F}$, there is a natural isomorphism $$ \text{Hom}_{A} \left( M , \Gamma(X, \mathcal{F}) \right) \simeq \text{Hom}_{\mathcal{O}_{X}} \left( \widetilde{M}, \mathcal{F} \right). $$ In other words, show that the global section functor and the twidlification functor are an adjoint pair. The bijection seems easy enough. It's the naturality that is making it hard. I really wanted to use Yoneda's lemma here, so I tried that but ended up coming to a conclusion that (I think?) is absurd. This was my attempt:
Fix a sheaf of $\mathcal{O}_{X}$-modules $\mathcal{F}$. Define the functors \begin{align} h_{\Gamma(X, \mathcal{F})} &:= \text{Hom}_{A} \left( -, \Gamma(X, \mathcal{F}) \right) \\ G(-) &:= \text{Hom}_{\mathcal{O}_{X}} \left( \widetilde{-} , \mathcal{F} \right) \end{align} Then Yoneda's lemma tells us that there is a natural isomorphism (as functors into the category of sets) $$ \text{Nat} \left( h_{\Gamma(X, \mathcal{F})} , G \right) \simeq \text{Hom}_{\mathcal{O}_{X}} \left( \Gamma(X, \mathcal{F})\tilde{\,} , \mathcal{F} \right). $$ This is where I became confused. The above line seems to be suggesting that there can only exist a natural isomorphism between the functors $h_{\Gamma(X, \mathcal{F})}$ and $G $ if $\mathcal{F}$ is quasi-coherent. But then the exercise asks to show that this adjunction holds in the entire ambient category of $\mathcal{O}_{X}$-modules.
My guess is that I have misunderstood how to apply Yoneda's lemma here. In fact, I used it here specifically because I wanted some practice with applying it. Can Yoneda's lemma be used for this exercise at all? If so, where have I gone wrong?
As an aside: Does anyone have some general advice for learning and understanding adjoint functors and Yoneda's lemma well, particularly being able to show that two functors are adjoint? This is one concept that I keep struggling with despite coming to grips with other content in algebraic geometry. I've added a tag for soft question for this last remark/question.