Reading some K-theory I understand the construction of the abelian group $K(X)$ where $X$ is a compact Hausdorff space. However, I read in most of the references that $K$ is also a functor. Why? Which are the categories that are being connected by $K$? What are their respective morphisms? This is confusing for me.
My thoughts: Since $$K:\mathrm{Vect}(X) \to \mathrm{Vect}(X) \times\mathrm{Vect}(X)\Big/\sim \;$$ is a mapping from an abelian monoid to an abelian group, $K$ would be a functor from the category of abelian monoids to the category of abelian groups. But an abelian group is also an abelian monoid. Does not this imply that $K$ is just a morphism? Then, which categories does the function $K$ join?
Many Thanks.
The categories are compact Hausdorff spaces with continuous maps, and Abelian groups with group homomorphisms. The functor is contravariant, and is induced via pullback of vector bundles.