I'm studying Lie Algebras from Humphreys's book. The definition of isomorphism of root systems seems to me to have some functor properties: if $\phi:(\Phi,E)\to(\Phi', E')$ is an isomorphism of euclidean spaces $E,E'$ and root systems $\Phi,\Phi'$, then for any root $\alpha\in\Phi$ we have $\sigma_{\phi(\alpha)}\circ \phi=\phi\circ\sigma_{\alpha}$.
My question is: Is there any way to see a root system as a category and an isomorphism between root systems as functor between categories? The idea is the same to think about a group (or a monoid, or a POSET, ecc) as a unique category.
For example, if we see the Weyl's groups as categories, $\phi$ defines a functor between them.
My idea is to use the reflections $\sigma_{\alpha}$ as morphisms and the roots as objects, but I think something is missing.