Is there a universal way to construct a bimonoidal category structure on $\mathsf{PSh}(\mathcal{C})$ from such a structure on $\mathcal{C}$? 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 $\mathcal{D}$?
 A: $\def\C{\mathcal{C}}\def\Set{\mathsf{Set}}$
(What follows comes from a private chat with Todd Trimble)
Notation. If $(\C,\otimes,\oplus)$ is a bimonoidal category, I will call $\otimes$ the multiplicative structure and $\oplus$ the additive structure; if $\oplus$ is the cocartesian monoidal structure, I will call $\C$ a 2-rig, following https://arxiv.org/abs/2103.00938.
I am convinced that in general (=for a general bimonoidal category) little can be said, because one needs some compatibility between the multiplicative structure and co/products. Even when $\C$ is a 2-rig the most I can formulate until now is a
Conjecture. When $\C$ is a 2-rig, the category $[\C^o,\Set]_\times$ of functors $\C^o \to \Set$ that are product-preserving (=sending coproducts in $\C$ to products in $\Set$) is the free 2-rig on $\C$.
In order for this freeness property to be legitimate, the least we can ask is that

*

*$F\C=[\C^o,\Set]_\times$ is a 2-rig if the multiplicative structure is Day convolution;

*The Yoneda embedding $y : \C \to F\C$ is a morphism of 2-rigs.

Unfortunately, I am still unable to prove that the Day convolution restricts to "models" of the "theory" $\C$ (it is a fruitful intuition to think of $\C$ like it was a Lawvere theory even if it's not, were it only because it's easier to query google with questions ;-) )
Update: I couldn't because it's not true, but it falls very short from being true, in the sense that the conjecture is "true up to reflecting the monoidal structure": observe that  $[\C^o,\Set]_\times$ has many desirable properties for the free cocomplete 2-rig on $\C$:

*

*$[\C^o,\Set]_\times$ is a cocomplete[¹], reflective subcategory of the entire $[\C^o,\Set]$.

*the yoneda embedding $y : \C \to [\C^o,\Set]$ clearly factors through $[\C^o,\Set]_\times$.

*$F\C=[\C^o,\Set]_\times$ has the following universal property, if $\mathcal D$ is cocomplete:
$$\{\text{cocontinuous } g : F\C \to\mathcal D\}\cong 
\{\text{coproduct preserving } h : \C \to \mathcal D\}$$

*Another useful universal property for $[\C^o,\Set]_\times$ is that it is the cocompletion of $\C$ under sifted colimit.

All these facts turn out to be useful to establish that $[\C^o,\Set]_\times$ is the free 2-rig over $\C$:  the coend that expresses the usual Day convolution formula is to be interpreted in a way that we first take the coend in the usual category of presheaves, but then to that apply the reflection functor $r$ that is left adjoint to the full inclusion. So, it's not true that the ordinary Day convolution takes a pair of product-preserving functors to a product-preserving functor; you have to sheafify. After you do that, everything falls into place.
Look how neat everything becomes!
Suppose we have a functor $F$ in $[\C^o,\Set]_\times$; by 4 above, it's a sifted colimit of representables $\C(-, c)$. Since we assume $\C$ is a 2-rig, we have a composite of (finite) coproduct-preserving functors
$$Y : \C \stackrel{C \otimes -}{\to} \C \stackrel{y}{\to} [\C^o,\Set]_\times$$
under the equivalence of point 3, this becomes a colimit preserving functor $[\C^o,\Set]_\times \to [\C^o,\Set]_\times$, exactly the reflected convolution $\C(-,C)\ast \_$.
But since the convolution we want to build must be cocontinuous, it is uniquely determined by this construction!
The missing detail is some result ensuring that $\ast$ is a monoidal structure. All in all I expect this to be a consequence of a theorem about transport of monoidal structures $\otimes$ (Day convolution of presheaves) into $\ast$ ("reflected" Day convolution) given some lax monoidality assumptions on $r$, so I won't enter the details.
Let me just add a final neat detail: the coend formula expressing the Day convolution product is a reflexive coequalizer; a reflexive coequalizer is a sifted colimit, so it can be interpreted as the usual, pointwise colimit in Set. Moreover, the sums = coproducts involved in the coend formula are filtered colimits of finite coproducts; again, filtered colimits are sifted colimits. So the only "re-interpretation" involved with the reflector resides in the way you compute finite coproducts in models, i.e.: you don't compute finite coproducts set-wise, but at the level of models, and every other colimit as usual.

[¹] But colimits in $[\C^o,\Set]_\times$ are not computed as colimits in the presheaf category (thikn again to the case of Lawvere theories and coproducts of monoids...), and this will be crucial, because for example, in $[\C^o,\Set]_\times$ the object $\C(-,A+B)$ has the universal property of the coproduct $\C(-,A)+\C(-,B)$.
