Unicity of Enrichment over Cat or Gpd

Enrichment of ordinary monads over Cat or Gpd is fundamental to Max Kelly’s unified theory of coherence for categories with structure. So here, we investigate existence and unicity of enrichments of ordinary func­ tors, natural transformations, and hence also monads, over Cat and Gpd. We show that every ordinary natural transformation between 2-functors whose domain 2-category has either tensors or cotensors with the arrow category is 2-natural. We use that to prove that an ordinary monad, or endofunctor, on such a 2-category has at most one enrichment over Cat or Gpd. We also describe a monad on Cat that has no enrichment. So enrichment over Cat is a non-trivial property of a monad rather than a structure that is additional to it. Finally, we present an example, due to Kelly, of V other than Cat or Gpd and an ordinary monad for which more than one enrichment over V exists, showing that our main theorem is specific to Cat and Gpd.