There is a double category MonCat whose objects are monoidal categories, whose horizontal arrows are lax monoidal functors, whose vertical arrows are colax monoidal functors, and whose 2-cells are generalized monoidal natural transformations. An analogous double category can be constructed involving the algebras for any 2-monad.
I've searched through all the papers on double categories linked to on that page, but can find no reference to this double category, even in the paper [GrP99], which includes a list of examples of double categories.
Furthermore, I cannot find the definition of a generalized monoidal natural transformation anywhere online. I can work out what the appropriate coherence diagrams are, but I'd still be interested in knowing whether the definition had been published anywhere.
The double category in question is mentioned in this post on the n-Category Cafe, but without much explanation.
Is the double category MonCat ever discussed in detail in a published format of some kind, or is it purely folklore?