Complexification as a functor I know very little category theory, but I found some good notes and I am confident that I can look up the definitions if something is unclear. That being said, Wikipedia mentions that complexification can be considered a functor, but doesn't elaborate and I don't know another reference. If someone can give a detailed explanation, that would be great.
 A: For a field $k$ there is a category ${\rm Vec}_k$ whose objects are $k$-vector spaces and whose maps are $k$-linear maps. Complexification is a way to turn $\mathbb R$-vector spaces into $\mathbb C$-vector spaces. This suggests that there is a relation between the categories ${\rm Vec}_{\mathbb R}$ and ${\rm Vec}_{\mathbb C}$.
Let's try to define a functor $F : {\rm Vec}_{\mathbb R} \to {\rm Vec}_{\mathbb C}$ using complexification. On objects it works as expected: for $V \in {\rm Vec}_{\mathbb R}$ we define $F(V) = V^{\mathbb C} = V\otimes_{\mathbb R} \mathbb C$. This is a complex vector space with $\mathbb C$-scalar multiplication given by
$$
 \lambda \cdot (v \otimes z) = v \otimes \lambda z.
$$
Now we need to define $F$ on the morphisms of ${\rm Vec}_{\mathbb R}$, a reasonable choice is to assign, for every $\mathbb R$-linear map $f : V \to W$ the map $F(f) = f \otimes_{\mathbb R} {\rm id}_{\mathbb C} : V^{\mathbb C} \to W^{\mathbb C}$. The map $F(f)$ is $\mathbb C$-linear.
You can easily check that $F$ satisfies the definition of a functor, for example if $f, g$ are composable $\mathbb R$-linear maps we have $$F(gf) = gf\otimes {\rm id}_{\mathbb C} = (g \otimes {\rm id}_{\mathbb C}) (f \otimes {\rm id}_{\mathbb C} ) = F(g)F(f)$$
and also $F({\rm id}_V) = {\rm id}_{F(V)}$.
If you want to read more about this kind of functors in general you should take a look at induction and restriction of scalars.
A: Consider the obvious functor assigning a real vector space to each complex vector space, then complexification is an adjunction and this is just a reformulation of the definition I proposed in this question (i.e. both characterizations are equivalent):
I had a look at the second chapter in Tom Leinster's Basic Category Theory and it provides all the necessary results for a detailed explanation. In particular, we can prove the equivalence based on the different formulations of the definition of adjunctions that are presented in that chapter.
