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Let $\mathcal{C}$ and $\mathcal{D}$ be categories with finite products, and consider them as monoidal categories in the obvious way.

  1. Every functor $\mathcal{C} \to \mathcal{D}$ can be canonically equipped with the structure of an oplax braided monoidal functor, because the universal properties give us the required morphisms $F 1 \to 1$ and $F (X \times Y) \to F X \times F Y$ in $\mathcal{D}$. Is this structure unique, or at least unique up to isomorphism of oplax braided monoidal functors?

  2. Every functor $\mathcal{C} \to \mathcal{D}$ that preserves finite products is canonically a strong braided monoidal functor. Again, is this structure unique, or at least unique up to isomorphism of strong braided monoidal functors?

I suspect the question basically boils down to the existence of strict braided monoidal automorphisms, but I haven't been able to find any or show that there are none.

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