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Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be a (weak) monoidal 2-functor between two strict monoidal 2-categories. Up to replacing $\mathcal{C}$ by an equivalent strict monoidal 2-category, can I always assume that the monoidal 2-functor $F$ is strict?

I've seen similar results in the literature, but they do modify both the source and the target of $F$, whereas I want to keep the target unchanged!

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  • $\begingroup$ By "strict monoidal 2-category" do you mean a "strictly monoidal strict 2-category", i.e. a monoid object in the category of 2-category. i.e. it has both strict composition in dimension 1 and a strictly associative and unital tensor product which is a strict 2-functor ? or part of the structure is assumed to be weak ? $\endgroup$
    – Simon Henry
    Commented Apr 9, 2021 at 15:19
  • $\begingroup$ That's a good point; I was really thinking about Gray monoids, i.e. the underlying 2-category is strict and the tensor product is cubical (and a bunch of natural transformations are identites). $\endgroup$
    – JeCl
    Commented Apr 11, 2021 at 5:21
  • $\begingroup$ Right, so strict monoid for the (pseudo-)Gray tensor product. My intuitive impression is that the result you are after is true for these, but might be false for stricit monoid in the sense of the cartesian monoidal structure, but I don't know a reference. I'll think about it. $\endgroup$
    – Simon Henry
    Commented Apr 11, 2021 at 13:05

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