Direct limit in the category of fields 
Is it true that the direct limit of a directed system in the category of fields is just the union of fields in the system with obvious maps?

I think its true but i am confused because i am new to category theory. Any ideas?
 A: Yes, this is correct.  If $\{K_\alpha\}$ is a directed system, we can show that $K = \bigcup_\alpha K_\alpha$ has the universal property of a direct limit: if $L$ is any field, then a collection of compatible morphisms $f_\alpha : K_\alpha \to L$ lifts uniquely to a morphism $f: K\to L$.
To see this, note that any $a\in K$ comes from some $a_\alpha \in K_\alpha$, so we must have $f(a) = f_\alpha (a_\alpha)$.  Once we have shown that this is really a well-defined (independent of $\alpha$) homomorphism of fields with the desired property, this gives us uniqueness. (Exercise: Fill in the details!)
One important note: It is easy to casually write $K = \bigcup_\alpha K_\alpha$, but we have to understand that this is a very specific kind of union, a directed union.  We cannot just throw together the various $K_\alpha$, since they are not given as part of a larger set.  Rather, we need to take the disjoint union and identify pairs that are compatible in the directed system.  (This is the direct limit in the category of sets.)
