Let $X$ be a scheme and $\mathcal{F},\mathcal{G}$ be two sheaves of $\mathcal{O}_X$-modules. I showed that the presheaf which assigns each open subset $U$ of $X$, $$ U \longmapsto \mathrm{Hom}_{\mathcal{O}_X}(\mathcal{F}|_U,\mathcal{G}|_U) $$ is a sheaf. Let us denote it by $\mathcal{H} = Hom_{\mathcal{O}_X}(\mathcal{F},\mathcal{G})$.
In [Liu, "Algebraic Geometry and Arithmetic Curves"], page 172, Ex 1.5 we need to prove the following assertion (and I quote):
Let $X = \operatorname{Spec}(A)$ be an affine scheme and $\mathcal{F}$ a quasi-coherent $\mathcal{O}_X$-module (as a sheaf). Then he canonical map $$ \mathrm{Hom}_{\mathcal{O}_X}(\mathcal{F},\mathcal{G}) \longrightarrow \mathrm{Hom}_A (\mathcal{F}(X),\mathcal{G}(X)) $$ is a bijection.
In the LHS we have a sheaf, in the RHS with an $A$-module. I suspect there is a mistake in the question and the author ment $$ \mathcal{H}(X) = \left( \mathrm{Hom}_{\mathcal{O}_X}(\mathcal{F},\mathcal{G}) \right)(X) \longrightarrow \mathrm{Hom}_A (\mathcal{F}(X),\mathcal{G}(X)) $$ which makes more sense.
To prove this or the corrected statement I need to use two facts:
- Affine scheme may be covered by principal open subsets $D(f) = \{ x \in X \mid f \notin x \}$ where $x = \mathfrak{p} \in \operatorname{Spec}(A)=X$ is a prime ideal.
- The $\mathcal{F}$ is quasi-coherent and thus for every prime ideal $x \in X = \operatorname{Spec}(A)$ the stalk $\mathcal{F}_x$ is the localization of $\mathcal{F}(X)$ at $x$, that is $(\mathcal{F}(X))_x$.