I will state the fundamental theorem of Galois theory to make things clear. Let $F/K$ be a finite dimensional Galois extension. Let $A$ be the set of all intermediate fields of $F/K$, and let $B$ be the set of all subgroups of $\operatorname{Aut}_KF$. Let $f^*:A\to B$ be the function defined by $f^*(E)=\operatorname{Aut}_EF$. Define the function $f_*:B\to A$ by letting $f_*(H)$ be the fixed field of $H$. Then the fundamental theorem of Galois theory says that:
(1) $M\subseteq N$ implies $f^*(M)\geq f^*(N)$, and $I\geq J$ implies $f_*(I)\subseteq f_*(J)$;
(2) $f^*f_*$ and $f_*f^*$ are the identity functions;
(3) That $M\subseteq N$ is a Galois extension implies $f^*(M)\trianglerighteq f^*(N)$, and $I\trianglerighteq J$ implies that $f_*(I)\subseteq f_*(J)$ is a Galois extension.
We can rephrase (1) and (2) by saying that $(f^*,f_*)$ is a pair of category isomorphisms (viewing the preordered sets $(A,\subseteq)$ and $(B,\geq)$ as categories). But how can we rephrase (1) - (3) simultaneously? I do not know how to view structures such as $(B,\trianglerighteq,\geq)$ as categories.
