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Increasingly harder question, but a reference for the first would be ok:

  1. Is the category of (symmetric?) monoidal categories closed for limits like products?

  2. Is it true that the underlying category is the limit of the underlying categories? Would be ok if you can build a "free" monoidal category out of a category.

  3. In case (1) is "no", along which diagrams you can take the limit? Seems like there are no problems in the product (just set everything component by component).

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It depends on what "the" category of monoidal categories means.

  • The category $\rm MonCat_s$ of monoidal categories and strict monoidal functors is monadic over $\rm Cat$, so it is complete with limits created in $\rm Cat$. In fact it is complete as a 2-category, having all $\rm Cat$-weighted limits created in $\rm Cat$ as well.
  • The category $\rm MonCat_p$ of monoidal categories and strong (a.k.a. pseudo) monoidal functors is not complete. For instance, if $I$ denotes the two-object contractible groupoid, with its essentially unique monoidal structure, and $1$ the terminal category with its unique monoidal structure, then both functors $1\rightrightarrows I$ are strong monoidal, but have no equalizer in $\rm MonCat_p$. However, $\rm MonCat_p$ does have PIE-limits, which as you noted do include all products (the "P"). This implies that it has pseudo-limits, and therefore is complete as a bicategory (has all "bilimits", i.e. 2-categorical limits up to equivalence).
  • The category $\rm MonCat_l$ of monoidal categories and lax monoidal functors (sometimes called simply "monoidal functors") is not even complete as a bicategory. However it does have an important class of limits known as rigged limits.
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Just for the sake of completeness, this can be done for infinity monoidal categories with the following trick.

Infinity monoidal categories are precisely cocartesian fibrations over $N(Fin_*)$, which are the same as cartesian fibrations over $N(Fin_*)^op$. It turns out that there is a model category of marked simplicial sets over $N(Fin_*)^op$ such that fibrant objects are precisely cartesian fibrations, and weak equivalences are the ones that induces equivalences on map spaces toward a fibrant object.

Marked simplicial sets are manifestly complete, and it is easy to see that the limit (in the category of fibrant objects) of fibrant objects is the fibrant replacement of the limit computed as marked simplicial sets.

It is not clear if this construction preserves the fact that our monoidal categories were not infinity but just categories: is still true that we get a monoidal category (not infinity)?

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    $\begingroup$ For this to be true you need to take homotopy limits. If you only care about product then this will not be problem, but for more general limits, typical fiber product or equalizer you will need homotopy limits. The same is true for orinary category, you need to consider pseudo-limits for the results to be true (but for product this is again not a problem) $\endgroup$
    – Simon Henry
    Commented Dec 13, 2019 at 19:53
  • $\begingroup$ So is it still true that ordinary monoidal categories are closed, say, for pullbacks and pushouts? $\endgroup$
    – Andrea Marino
    Commented Dec 21, 2019 at 2:56
  • $\begingroup$ Only under pseudo-pullback, see Mike's answer. $\endgroup$
    – Simon Henry
    Commented Dec 21, 2019 at 9:11

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