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A new, self-contained, proof of a coherence result for categories equipped with two symmetric monoidal structures bridged by a natural transformation is given. It is shown that this coherence result is sufficient for ω × ω-indexed family of iterated reduced bar constructions based on such a category.Mathematics Subject Classification (2010): 18D10, 57T30, 03F07, 55P47

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.

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