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The question is in the title. I'm also happy to get answers about (your favorite adjective) monoidal categories.

Here's a guess:

In order to compute a colimit of monoids we can push everything down to Sets, compute the colimit there, freely generate a monoid from that, and add all the equations that held in any of the original monoids.

Can we do something similar here: Push everything down to Cat, compute the colimit there, freely generate a monoidal category from that, and add in equations/isomorphisms aligning the new monoidal product with the original ones.

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  • $\begingroup$ Strict-monoidal categories are just monoidal objects (i.e. models of the alegebrai theory "Mon" i.e."monoids") on the cartesian closed (meta)category $(CAT, \times)$. Form 6.42 p. 192 of P. Johstone "TOpos THeory" 1978: the forgetful functor $U: Mon(CAT)\to CAT$ create reflexive coeaqualizers. From 0.16 p.3-4 (some book) you get small colimts. FOr the coproduct the $\endgroup$
    – Buschi Sergio
    Commented Apr 6, 2015 at 19:56
  • $\begingroup$ I believe non-strict monoidal categories can be described as algebras over a monad which is a cofibrant replacement of the monad presenting strict monoidal categories. So a similar statement should hold. $\endgroup$
    – AAK
    Commented Apr 6, 2015 at 20:39
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    $\begingroup$ It might be useful to specify whether you want strict, weak, or lax colimits. $\endgroup$
    – André Henriques
    Commented Apr 6, 2015 at 21:05
  • $\begingroup$ Already for colimits of categories there's an issue of what kind of colimits you want. The naive notion (a colimit in the category of categories and functors) is not invariant with respect to equivalence of categories. To fix this one can use a notion of $2$-colimit in the $2$-category of categories, functors, and natural transformations (and one can decide whether natural transformations must be invertible as well). For monoidal categories there are three levels instead of two: we can work in a category, $2$-category, or $3$-category of monoidal categories... $\endgroup$
    – Qiaochu Yuan
    Commented Apr 6, 2015 at 23:31

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