Characterization of comma objects and Kan extensions preserved by pasting in MonCat

Ascertain whether the Kan extensions in MonCat that are preserved by pasting with comma squares are exactly those arising from the described construction, and determine whether the comma squares considered exhaust all comma objects that exist in the 2-category MonCat of monoidal categories, lax monoidal functors, and monoidal natural transformations.

Background

Subsection 5.3 introduces exactly pointwise monoidal codensity monads and discusses pointwise Kan extensions in MonCat via pasting with comma squares. It shows that certain exactly pointwise monoidal Kan extensions are preserved by pasting comma squares constructed from strong monoidal functors and lax monoidal functors.

The remark highlights that, while these examples work, it is unresolved whether they characterize all such preserved Kan extensions and whether these comma squares account for all comma objects in MonCat.