What is meant by a category of unramified extension of $K$ (a local field)? I am studying local fields and I came across this following result:
"Fix a local field $K$ with perfect residue field $k$. Then there is an equivalence of categories between the extensions of $k$ and the unramified extensions of $K$."
I am struggling to understand what the categories are here. Could someone please explain me what the objects and arrows are here?
 A: I rather agree with @LordSharktheUnknown. In a course on Galois theory for beginners, when working with several extensions of a given base field $K$, people are generally cautious to warn that these are contained in some larger extension of $K$. Given $K$, a more "functorial" setting (not useful to beginners) should be the category of $K$-extensions: the objects are fields $L$ with a (necessarily injective) field homomorphism $f:K \to L$, the morphisms between two objects $(L,f)$ and $(L', f')$ are field homomorphisms $g:L \to L'$ such that $g.f=f'$; such a morphism will be called "$K$-morphism" for short.
Back to the OP question, where we must relate two categories of extensions: the unramified extensions of a local field $K$, and the extensions of its (perfect) residual field $k$. Although the definition of a "local field" is not given, I suppose that, as usual, $K$ is the field of fractions of a complete dicrete valuation ring. The answer is given in full generality and detail in Serre's "Local Fields", chap.III, §5, thm. 3 : Let $L/K$ be finite unramified, with residual extension $l/k$, and let $L'/K$ be any finite extension, with residual extension $l'/k$ . Then the set of $K$-morphisms from $L$ to $L'$ corresponds bijectively (under reduction) to the set of $k$-morphisms from $l$ to $l'$. The proof relies on Hensel's lemma.
