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Let $MonCat$ and $Cat$ denote the 2-categories of monoidal categories with strict monoidal functors and small categories, respectively.

There is a forgetful functor $Forget:MonCat\rightarrow Cat$ Does this preserve (filtered) colimits? I am thinking of 2-colimits in MonCat.

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Yes. Intuitively, this is because all the operations of a monoidal category are "finitary".

One way to prove this formally is that the 2-monad for monoidal categories can be given a presentation in the category of finitary 2-monads, hence it is finitary (which is equivaent to its forgetful functor preserving filtered colimits). A good reference for presentations of 2-monads is Steve Lack's A 2-categories companion.

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