Symmetric monoidal extensions and graph cobordisms between finite sets

Given a symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ and a space $X$, we address the problem of explicitly describing the symmetric monoidal $(\infty,n)$-category freely obtained from $\mathcal{C}$ by adjoining $X$ new $n$-morphisms with prescribed sources and targets. We develop an apparatus of tools that allow one to detect in concrete situations such a free symmetric monoidal extension. As motivating application, we introduce a symmetric monoidal $(\infty,2)$-category ${\mathbb{G}\mathrm{r}}$ of graph cobordisms between finite sets, following classical constructions of Gersten, Culler--Vogtmann and Hatcher--Vogtmann, and we exhibit it as an extension of the symmetric monoidal $(\infty,1)$-category $\mathrm{Fin}$ of finite sets, obtained by freely adjoining a specific list of new 1-morphisms and 2-morphisms. We recover results of Barkan--Steinebrunner and of Galatius.