The Day convolution monoidal category structure $(\mathsf{PSh}(\mathcal{C}),\circledast,\mathsf{h}_{\mathbf{1}_{\mathcal{C}}})$ on the category of presheaves of a monoidal category $(\mathcal{C},\otimes,\mathbf{1}_{\mathcal{C}})$ satisfies the following universal properties:

- In
*A universal property of the convolution monoidal structure*, Im–Kelly prove that it is the free monoidal cocompletion of $\mathcal{C}$. Here, a monoidal category is**monoidally cocomplete**if it is cocomplete and the functors $A\otimes-$ and $-\otimes B$ preserve colimits for all $A,B\in\mathrm{Obj}(\mathcal{C})$. Im–Kelly then prove:

The monoidal category $(\mathsf{PSh}(\mathcal{C}),\circledast,\mathsf{h}_{\mathbf{1}_{\mathcal{C}}})$ is the universal monoidally cocomplete category on $\mathcal{C}$ in that, given any monoidally cocomplete monoidal category $(\mathcal{D},\otimes,\mathbf{1}_{\mathcal{D}})$, precomposition with $よ\colon\mathcal{C}\hookrightarrow\mathsf{PSh}(\mathcal{C})$ defines an equivalence of categories $$よ^*\colon\mathsf{Fun}^{\otimes,\mathsf{strong}}_{\mathsf{cocont.}}(\mathsf{PSh}(\mathcal{C}),\mathcal{D})\longrightarrow\mathsf{Fun}^{\otimes,\mathsf{strong}}(\mathcal{C},\mathcal{D}).$$ That is, $(\mathsf{PSh}(\mathcal{C}),\circledast,\mathsf{h}_{\mathbf{1}_{\mathcal{C}}})$ is uniquely determined by the following requirements:

- The Yoneda embedding $よ\colon\mathcal{C}\hookrightarrow\mathsf{PSh}(\mathcal{C})$ is strong monoidal.
- $\circledast$ is cocontinuous in each variable.

- Given another monoidal category $(\mathcal{D},\otimes_{\mathcal{D}},\mathbf{1}_{\mathbf{D}})$, we have an equivalence of categories $$ \left\{ \begin{gathered} \text{symmetric strong monoidal}\\ \text{functors $\mathcal{C}\times\mathcal{D}\to\mathsf{Sets}$} \end{gathered} \right\} \cong \left\{ \begin{gathered} \text{symmetric strong monoidal}\\ \text{functors $\mathcal{D}\to\mathsf{PSh}(\mathcal{C})$} \end{gathered} \right\}, $$ natural in $\mathcal{D}$.

The analogue of the first of these for bimonoidal categories, however, doesn't work.

**Question.** Given a bimonoidal category $(\mathcal{C},\otimes,\oplus,\mathbf{1}_{\mathcal{C}},\mathbf{0}_{\mathcal{C}})$, is there a universal way to put a bimonoidal category structure on $\mathsf{PSh}(\mathcal{C})$?

In particular, is there a bimonoidal category structure on $\mathsf{PSh}(\mathcal{C})$ such that, given another bimonoidal category $(\mathcal{D},\otimes_{\mathcal{D}},\oplus_{\mathcal{D}},\mathbf{1}_{\mathcal{D}},\mathbf{0}_{\mathcal{D}})$, we have an equivalence of categories $$ \left\{ \begin{gathered} \text{symmetric strong }\color{red}{\text{bi}}\text{monoidal}\\ \text{functors $\mathcal{C}\times\mathcal{D}\to\mathsf{Sets}$} \end{gathered} \right\} \cong \left\{ \begin{gathered} \text{symmetric strong }\color{red}{\text{bi}}\text{monoidal}\\ \text{functors $\mathcal{D}\to\mathsf{PSh}(\mathcal{C})$} \end{gathered} \right\}, $$ natural in $\mathcal{D}$?