We define a functor $\Phi \colon \mathcal{O}_G^\mathrm{op} \to \textup{\textsf{cat}}_F^E$ that sends $H \subset G$ to the subfield of $E$ of elements that are fixed by $H$ under the action of the Galois group. If $G/H \to G/K$ is induced by $\gamma$, then the field homomorphism $x \mapsto \gamma x$ sends an element $x \in E$ that is fixed by $K$ to an element $\gamma x \in E$ that is fixed by $H$. This defines the action of the functor $\Phi$ on morphisms. The fundamental theorem of Galois theory asserts that $\Phi$ defines a bijection on objects but in fact more is true: $\Phi$ defines an isomorphism of categories $\mathcal{O}_G^\mathrm{op} \cong \textup{\textsf{cat}}_F^E$.