As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such that $f(\Phi) \subseteq \Phi'$ and $f(\alpha) \dashv f(\beta) = \alpha \dashv \beta$ for all $\alpha, \beta \in \Phi$, where $\alpha \dashv \beta \mathrel{:=} 2 \frac{\langle \alpha | \beta \rangle}{\langle \beta | \beta \rangle} \in \mathbb{Z}$ and $\langle ~ |~ \rangle$ denotes the inner product on $E$. For isomorphisms of root systems, this last condition is in fact redundant, and follows automatically from the first two. This seems to be the "correct" notion of morphism for root systems: we have finite coproducts of root systems (which are oddly denoted by a product in most texts) and an initial object (the empty root system). Moreover, the Weyl group construction is a covariant functor on root systems.
Now, given a complex semisimple Lie algebra $\mathfrak{g}$, we can assign to it a root system $\Phi$, given by the set of nonzero weights of the representation of a Cartan subalgebra $\mathfrak{h} \leq \mathfrak{g}$ on $\mathfrak{g}$ via the adjoint action.
This assignment of a root system to a complex semisimple Lie algebra is an invariant, in the sense that isomorphic Lie algebras have isomorphic root systems (and conversely). But is this construction functorial? That is given a morphism of semisimple Lie algebras $f: \mathfrak{g} \to \mathfrak{g}'$, do we obtain a morphism of their root systems? If not a covariant functor, might this be a contravariant functor?
I am just learning about the classification of Lie algebras now and do not have a great handle on the material yet, but I have a feeling that this should be the case.
One point that was raised by Mariano Suárez-Alvarez in the comments is that the choice of Cartan subalgebra is not functorial. Nevertheless, in the end the construction of a root system does not depend on the choice of Cartan subalgebra, which leaves me hope that this construction might still be functorial.
If it is indeed functorial, then the classification of semisimple complex Lie algebras would have an elegant description in terms of this "root system" functor: it is strong monoidal, conservative, and reflects simple objects (irreducible root systems being the simple objects in the category of root systems). Furthermore, since the Weyl group of a root system is functorial, then the assignment of the Weyl group to a Lie algebra would simply be the composite of the two functors.