Universal property of graph cobordisms

We exhibit the symmetric monoidal $\mathrm{\infty }$-category $\mathrm{G}\mathrm{r}\mathrm{C}\mathrm{o}\mathrm{b}$ of graph cobordisms between spaces as a full $\mathrm{\infty }$-subcategory of the $\mathrm{\infty }$-category $\mathrm{P}\mathrm{s}\mathrm{h}(\mathrm{G}\mathrm{r})$ of presheaves over $\mathrm{G}\mathrm{r}$, where $\mathrm{G}\mathrm{r}$ is the symmetric monoidal $\mathrm{\infty }$-category of graph cobordisms between finite sets. We also describe a universal property for $\mathrm{G}\mathrm{r}\mathrm{C}\mathrm{o}\mathrm{b}$: it is, in a suitable sense, the free symmetric monoidal extension of $\mathrm{G}\mathrm{r}$ endowed with the factorization homology ${\int }_X\ast$ of the universal ${\mathbf{E}}_{\mathrm{\infty }}$-Frobenius algebra for all spaces $X$. As a corollary, we identify the space of universal natural operations on the factorization homologies, in particular the Hochschild homology, of ${\mathbf{E}}_{\mathrm{\infty }}$-Frobenius algebras.