Completeness of 2-category of Monoidal Categories Is the 2-category of monoidal categories complete?
If not, can any conditions be imposed to satisfy completeness?
 A: When working with 2-categories, the notion of completeness is not quite as clear-cut as in 1-categories. See the nlab for more information. It should be relatively straightforward to check directly that monoidal categories have all 2-limits, though, and that the forgetful functor to $\mathsf{Cat}$ preserves them. For instance, the product of some monoidal categories is their product as in $\mathsf{Cat}$, with the monoidal product defined in the obvious way. A systematic way to see this is to note that monoidal categories are the pseudoalgebras for the "free monoid" 2-monad on the 2-category $\mathsf{Cat}$, and Blackwell, Kelly, and Power showed that such 2-categories always have all 2-limits.
The intuition is similar to constructing limits of algebras for a 1-monad on $\mathsf{Set}$, say. In this setting, it's pretty intuitive to work out that to create a limit in $\mathsf{Grp}$ or $\mathsf{Ring}$ or $\mathsf{Vect}$, say, you simply take the limit of the underlying sets and then there's really only one way you could possibly define the operations -- everything just works. This is abstracted to argue that the category of algebras for a 1-monad has limits created in the base category. Then the argument for 2-monads is really just the same thing one category level higher - modulo strictness issues.
EDIT
As Kevin Carlson  points out below, I cut too many corners in the above. In order to use the BKP result, you have to know that the monoidal categories are actually the strict algebras for a 2-monad with rank, which is also true, but the 2-monad in question is not quite as snappy to desribe as the "free monoid monad" $T$. Rather, It's the 2-monad $T'$ that freely adjoins all the structure of a monoidal category, including images of the $\otimes$ functor as well as associator morphisms, etc. all subject to certain equations of morphisms (but not of objects!).
But is there a conceptual way that we can say that the free monoidal category monad $T'$ is a "corrected" version of the free monoid monad $T$? Indeed there is. Steve Lack constructs a 2-model structure (i.e. a $\mathsf{Cat}$-enriched model structure) on the 2-category of finitary monads on $\mathsf{Cat}$ (or any other reasonable 2-category). This model structure is lifted from a trivial model structure on the 2-category of endo-2-functors of $\mathsf{Cat}$. And in this model structure, $T'$ is a cofibrant replacement for $T$. And in general the 2-category of strict algebras for the cofibrant replacement is biequivalent to the 2-category of pseudoalgebras for the original 2-monad (although to be honest I don't quite see why this should always be the case on general grounds).
