Schur functors and categorified plethysm

It is known that the Grothendieck group of the category of Schur functors is the ring of symmetric functions. This ring has a rich structure, much of which is encapsulated in the fact that it is a plethory: a monoid in the category of birings with its substitution monoidal structure. We show that similarly the category of Schur functors is a 2-plethory, which descends to give the plethory structure on symmetric functions. Thus, much of the structure of symmetric functions exists at a higher level in the category of Schur functors.