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A monoidal category is a bicategory with one object. So from this perspective, one would be tempted to define a tricategorie $ \overline{\mathbf{Mon Cat}} $ of monoidal categories with

  • lax functors between one-object bicategories as 1-morphisms,
  • lax transformations between such functors as 2-morphisms,
  • modifications between such transformations as 3-morphisms.

But of course according to the usual definition $ \mathbf{Mon Cat} $ constitutes just a bicategory with 1- and 2-morphisms being monoidal functors and monoidal trafos.

In fact these two definitions still agree on the 1-morphisms. I.e. a lax functor of one-object-bicategories corresponds precisely to a lax functor of monoidal categories. They don't agree on the 2-morphisms though:
Among other things, a 2-trafo $\sigma: F \to G: \mathbf{A} \to \mathbf{B} $ carries objects $a \in A$ to morphisms $\sigma_a \in \mathbf{B}(Fa,Ga)$ and nothing in the definition of monoidal trafo corresponds to that.
Note that a similar problem arises also on the level of 1 morphisms, but there the part of a 2-functor which carries objects to objects becomes trivial, since the relevant bicategories just have one object.

Long story short, it seems to me when considering $\mathbf{MonCat}$ it is a bit misleading to think of monoidal categories as one object bicategories, because the monoidal functors and trafos are really more like special 1-categorical functors and trafos, than like 2-categorical functors and trafos.

My question is now, is $\mathbf{MonCat}$ (seen as a tricategory with idendity 3-cells) a subtricategory of $\overline{\mathbf{MonCat}}$? Or is there any other interesting relation, between these two different possibilities to interpret monoidal categories as objects in higher categories? And although this seems to be a bit of a soft question, is there a good conceptional reason, why $\mathbf{MonCat}$ is defined the way it is?
Regarding the last point, I would appreciate an argument which generalizes to the correct definition of the tricategory of monoidal bicategories. (Where completely analog to above the fact that monoidal bicategories are just one objects tricategories will immediately yield a (weak) 4-category, but not the desired tricategory I guess.)

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  • $\begingroup$ A monoidal transformation does have something corresponding to the 2-transformation's action on objects, the unit map of the monoidal transformation. If you have a monoidal transformation $(\sigma,m):F\to G$ where $m$ is the unitor (I forget what the correct name is) map, thinking of things in terms of bicategories with a single object $\star$, you can think of the map $m$ as the image of $\star$, $\sigma(\star)=m:F(1)\to G(1)$. $\endgroup$ – Dom Feb 26 '18 at 10:18
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I asked the $n-1$ version of this question here, and got a good answer. So I suspect you can categorify that. I'm afraid I can't fill in the details since I get a bit lost at the level of modifications.

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