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We often describe a category by giving a (directed, multi-)graph and freely generating a category of paths. I would like to know to what degree this intuition generalizes to monoidal categories, and where I can find details of such a construction. I am interested in both the symmetric and non-symmetric cases, so arguments/references for either are welcome.

Specifically, I would like to specify some atomic objects and atomic maps and use these to generate a monoidal category. In particular, the atomic maps should be able to map in or out of monoidal products, so we can't just go from graph to category to monoidal category.

I suspect that the "right" answer involves looking at graphs over the free monoid on the set of atomic objects. One could then produce a category and then a monoidal category from this. However, then one would need to align the monoid product on atomic objects with the monoidal product in the category, and I'm not sure exactly how that should work.

All of the above ignores questions of strictness. I am not sure yet what the appropriate context for my application is, so answers regarding any level of strictness are welcome.

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