Let $MonCat$ denote the 2-category of monoidal categories and strict monoidal functors. Let $Cat$ denote the category of 2-categories. There is a forgetful functor $Forget:MonCat\rightarrow Cat$. Does this have a left adjoint?
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$\begingroup$ Sorry I misread your question: you said that cat is the category of $2$-categories, that you meant that $Cat$ is the $2$-category of categories? And by $MonCat$ you mean the category of strict-monoidal-categories? Because in that case I'm not entirely sure that my answer still does apply. $\endgroup$– Giorgio MossaJul 29, 2016 at 19:08
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1 Answer
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Short answer: yes.
A little longer answer: you can apply theorem 1 in the following link, using the fact that $MonCat$ is the category of monoids in $Cat$, that $Cat$ is monoidal (with respect to the cartesian product), it has countable coproducts and that products distributes with coproducts.
Hope this helps.
answered Jul 29, 2016 at 13:43
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$\begingroup$ MonCat is a category of pseudomonoids in Cat. So this construction lands in strict monoidal categories-which is perfectly fine. $\endgroup$ Jul 29, 2016 at 16:37
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$\begingroup$ @KevinCarlson Thanks for pointing out.... Actually I'm not so sure that the argument works for non-strict categories. Since the OP was interested in monoidal categorie and strict functors I've assumed that always was strict.... maybe I was wrong........ $\endgroup$ Jul 29, 2016 at 19:13