Consider $\mathscr{F}$ to be a module of finite presentation over a ringed space $X$. I want to prove that, for any sheaf of modules $\mathscr{G}$ and $x \in X$, the canonical morphism $\mathscr{H}om(\mathscr{F}, \mathscr{G})_x \to \operatorname{Hom}_{\mathscr{O}_{X,x}}(\mathscr{F}_x, \mathscr{G}_x)$ is iso. To do so, I am following Görtz-Wedhorn's book, proposition 7.27. In the book, they consider two contravariant functors which they define by: $$\mathcal{F}: \mathscr{F} \mapsto \mathscr{H}om(\mathscr{F}, \mathscr{G})_x$$ $$\mathcal{F}': \mathscr{F} \mapsto \operatorname{Hom}_{\mathscr{O}_{X, x}}(\mathscr{F}_x, \mathscr{G}_x)$$
Now, I am somewhat confused as to what this contravariant functor mean. Is $\mathcal{F}$ meant to be the functor $\mathscr{H}om(\mathscr{F}, -)_x$, and $\mathcal{F}'$ the functor $\operatorname{Hom}_{\mathscr{O}_{X, x}}(\mathscr{F}_x, -_x)$? If so, how are these functors contravariant?