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From the nLab article on double categories:

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?

[GrP99]: Grandis, Marco; Paré, Robert, Limits in double categories, Cah. Topologie Géom. Différ. Catég. 40, No.3, 162-220 (1999). ZBL0939.18007.

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  • $\begingroup$ See if this helps you. See also here, sir. $\endgroup$ – Rohan Jan 7 '18 at 14:59
  • $\begingroup$ @Rohan Thanks, but these seem to be about monoidal double categories. I am asking about the (plain) double category whose objects are monoidal categories. $\endgroup$ – John Gowers Jan 7 '18 at 17:30
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    $\begingroup$ I think it's the double category Mcat defined in "Adjoints for double categories", Section 2.3, no? It's defined as the full sub double category of Dbl, the double category of double categories, on one-object double categories, so you'll have to check out the definition of that to get a definition of generalised monoidal natural transformations, but the coherence conditions are spelled out. $\endgroup$ – Tom Hirschowitz Jan 8 '18 at 8:27
  • $\begingroup$ Thank you, @Tom, that's what I was looking for. I remember now that I skipped over that paper because I thought that any examples of double categories would have been in the first part. $\endgroup$ – John Gowers Jan 9 '18 at 1:34
  • $\begingroup$ If you want the bounty, then just post your comment as an answer $\endgroup$ – John Gowers Jan 9 '18 at 1:34
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I think it's the double category Mcat defined in "Adjoints for double categories", Section 2.3. It's defined as the full sub double category of Dbl, the double category of double categories, on one-object double categories, so you'll have to check out the definition of that to get a definition of generalised monoidal natural transformations, but the coherence conditions are spelled out.

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