Let $\cal J = \bf Cat$ be the strict 2-category of small categories, functors and natural transformations and $\mathbb{V} : {\cal J}°\to \bf MonCAT$ a strict 2-functor taking value on possibly large monoidal categories, i.e. such that $\mathbb V(I)$ is a monoidal category for each $I\in\cal J°$.

This gadget is a monoidal prederivator, as defined in here (caveat: the nLab has a definition too, inspired by a different -higher- $n$POV; I do not consider that!)

Let now $\mathbb X$ be any strict 2-functor ${\cal J}°\to {\bf CAT}$. Consider the functor $$ [\![\mathbb X,\mathbb V]\!] : J\mapsto {\bf PsdNat}[\mathbb X,\mathbb V^J] $$ (the large category of pseudonatural transformations between $\mathbb X$ and the shifted prederivator $\mathbb V^J\colon I\mapsto \mathbb V(I\times J)$.

This works as an internal hom in the 2-category of prederivators, pseudonatural transformations, and modifications.

It defines a new strict 2-functor ${\cal J}\to \bf MonCat$. I would like to call this the pontwise monoidal structure on $[\![\mathbb X,\mathbb V]\!]$: aas a matter of fact, it's easy to prove that each category $[\![\mathbb X,\mathbb V]\!](J)$ is monoidal: given two objects $P,Q$ in it, we define $$ P\otimes Q : J\mapsto P_I\otimes_{I\times J}Q_I$$ to be the composition $$ \mathbb{X}(I)\xrightarrow{\Delta} \mathbb{X}(I)\times\mathbb X(I) \xrightarrow{P_I\times Q_I} \mathbb{V}^J(I)\times \mathbb{V}^J(I) \xrightarrow{\otimes_{I\times J}}\mathbb{V}^J(I)$$ This is a vertical composition of pseudonatural transformations, so it's pseudonatural in $I$, and apparently defines a monoidal structure.

Building on this, I would like to define the analogue of the Day convolution in the setting of (pre)derivators. Let me recall that given a small symmetric monoidal category $(C, \oplus)$ and presheaves $P,Q : C \to \bf Set$ we can define $$ P * Q : c\mapsto \int^{xy}Px\times Qy\times C(c, x\oplus y) $$ (of course you can do something similar in enriched setting); in a beautiful paper (Im-Kelly, "A universal property of the convolution monoidal structure") it is proved that $\widehat{C}^\otimes = ([C,{\bf Set}],*)$ is the free monoidal cocompletion of $C$, and it is the universal category rendering the Yoneda embedding $y : C \to \widehat{C}^\otimes$ a strong monoidal functor (moreover, $\widehat{C}^\otimes$ is monoidal closed; this is less interesting for the moment). So, here's the question:

Let $\mathbb X$ be small, monoidal (with respect to $\oplus$) and $\mathbb V$-enriched: this means that each $\mathbb X(I)$ is a small $\mathbb V(I)$-category and a bunch of compatibility conditions are satisfied, and moreover each $\mathbb X(I)$ is monoidal wrt $\oplus_I$. Is there a way to define a convolution monoidal product rendering the Yoneda morphism $$ y : (\mathbb X^\text{op},\oplus) \xrightarrow{\qquad} ([\![\mathbb X,\mathbb V]\!],*) $$ a "strong monoidal" one (see again Groth for the definition of monoidal morphism)?


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