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Let $(\mathcal V,\otimes,I)$ be a closed symmetric monoidal category, and let $\mathcal C$ be a $\mathcal V$-enriched category. The (weak) enriched Yoneda Lemma gives us a nice description of the set $Hom(F,G)$ of natural transformations between two $\mathcal V$-enriched functors $F,G\colon\mathcal C\to\mathcal V$ when $F$ is representable: it is in bijection with the set of maps $I \to G(Y)$ in $\mathcal V$ where $Y$ is an object representing $F$.

Now suppose that $\mathcal C$ itself is a monoidal category, and that our two functors $F$ and $G$ are monoidal functors. Is there a similarly nice description of the set $Hom^\otimes(F,G)$ of monoidal natural transformations between the functors, again in the case that $F$ is representable?

My suspicion is that the following might be true (possibly with extra conditions on $\mathcal C$). The fact that $F$ is a (lax) monoidal functor induces the structure of a comonoid on the representing object $Y$, and so there is an induced comonoid structure on $G(Y)$. My guess would be that monoidal natural transformations $F\to G$ are in bijection with morphisms of comonoids $I\to G(Y)$, but I can't prove this in general. (I can prove this in the case that $\mathcal V$ is the category of sets with cartesian product, but only for trivial reasons: every map in Set is a morphism of comonoids, and every natural transformation between monoidal Set-valued functors is a monoidal transformation.)

I would be especially interested in any references where this might be addressed.

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  • $\begingroup$ Your suspicion is correct; one way to see it with few computations is using the Day convolution monoidal structure on $Fun_V(C,V)$ (if $C$ is small and $V$ cocomplete). However, I don't know a reference (I know references for Day convolution but not for this specific statement) $\endgroup$ Apr 24 at 16:18
  • $\begingroup$ sciencedirect.com/science/article/pii/0022404986900058 this is a standard reference for the universal property of the Day convolution $\endgroup$
    – fosco
    Apr 24 at 21:10
  • $\begingroup$ @Alexander I've added a more streamlined result to my answer below. This is a nice result that, to me, follows straightforwardly from "formal category theory for monoidal categories", in the sense of my paper, and also shows the usefulness of that approach. Of course it can be proved directly as well. Let me know if you have any questions. $\endgroup$ Apr 27 at 12:40
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For your suspicion to work you need $G$ to be pseudomonoidal I would think, otherwise I don't see how to obtain a comonoid structure on $G(Y)$ from that on $Y$?

With both $F$ and $G$ lax monoidal considering Day-convolution $\hat\oslash$ on $\hat A = \mathcal V^{A^\text{op}}$ gives you a "monoidal yoneda lemma", not as nice as you suspected, as follows. The universal property of Day-convolution is that it creates the yoneda embedding $\text y\colon A \to \hat A$ in the double category of monoidal profunctors $A \nrightarrow B$, that is lax monoidal functors $J\colon A^\text{op} \otimes B \to \mathcal V$, and lax monoidal functors (and you really want a double category here I believe, a 2-category won't work or at least not as nicely). Other properties such as it defining $\hat A$ as the monoidal free cocompletion follow from that. The property that we will use is that it induces an equivalence of between the category of monoidal profunctors $A \nrightarrow B$ and that of lax monoidal functors $B \to \hat A$, mapping $J\colon A \nrightarrow B$ to any chosen $J^\lambda\colon B \to \hat A$ such that $J \cong \hat A(\text y, J^\lambda)$ as monoidal profunctors (e.g. take $J^\lambda(y) = J(-, y)$.

Writing $I$ for the unit $\mathcal V$-category with its monoidal structure, lax monoidal functors $A \to \mathcal V$ are precisely monoidal profunctors $A \nrightarrow I$. Your functor $F\colon A^\text{op} \to \mathcal V$ being representable means that there is a lax monoidal functor $y\colon I \to A$, i.e. a monoid $y \in A$, such that $F$ corresponds to the monoidal profunctor $A(-, y)$. Under the equivalence the latter corresponds to $A(-, y)^\lambda = \text yy\colon I \to \hat A$. Monoidal transformations $F \Rightarrow G$ are then transformations $A(-, y) \Rightarrow G$ of monoidal profunctors $A \nrightarrow I$ which, by the equivalence, correspond precisely to monoidal transformations $\text yy \Rightarrow G^\lambda$ of lax monoidal functors $I \to \hat A$. Since $G \cong \hat A(\text y, G^\lambda)$ the latter can be thought of as given by a $\mathcal V$-morphism $\phi\colon I \to G(y)$, but its compatibility with the monoidal structures has to be written as a commuting diagram of two parallel morphisms $I \otimes I \to \hat A(\text yy \hat\oslash \text yy, G^\lambda)$. Only when $F$, and hence $y$, is pseudomonoidal the previous isomorphism restricts to $G(y) \cong \hat A(\text yy, G^\lambda)$ of monoidal profunctors $I \nrightarrow I$, and the condition on $\phi$ reduces to it being a morphism of monoids.

I did not think much about $G$ being pseudomonoidal. You might want to look at the double category of monoidal profunctors and colax monoidal functors instead?

As for references, the equivalence between monoidal profunctors and lax monoidal functors can be extracted from Section 2 of Pisani's Sequential Multicategories (although at the moment I don't remember exactly how...) The bigger picture of Day-convolution creating monoidal yoneda embeddings is one of the main motivations of my draft paper A double-dimensional approach to formal category theory

Edit: Thinking a bit more about this, using the fact that $\text y$ is pseudomonoidal allows us to streamline the condition on $\phi$ above further. All put together we get:

Monoidal Yoneda lemma. Let $G\colon A^{\text op} \to \mathcal V$ be lax monoidal and $y \in A$ a monoid, thus making $A(-,y)\colon A^{\text op} \to \mathcal V$ lax monoidal. Monoidal transformations $A(-, y) \Rightarrow G$ correspond precisely to morphisms $I \to Gy$ in $\mathcal V$ such that the composites $$ I \otimes I \to Gy \otimes Gy \xrightarrow{G_\otimes} G(y \otimes y) \qquad \text{and} \qquad I \otimes I \cong I \to Gy \xrightarrow{Gy_\otimes} G(y \otimes y)$$ coincide, as well as (as Alexander points out below) $$ I \to Gy \xrightarrow{G(y_I)} G(I_A) \qquad \text{and} \qquad I \xrightarrow{G_I} G(I_A) $$ where $G_I$ and $y_I$ are the unitors of $G$ and $y$.

If $G$ is pseudomonoidal then this recovers the lemma that you suspected and Maxime proved.

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  • $\begingroup$ I think $G$ is assumed to be strong monoidal $\endgroup$ Apr 26 at 15:31
  • $\begingroup$ @MaximeRamzi Well not explicitly, only for the suspicion to work. Since the first thing you find here is that F being pseudomonoidal is not useful, it seems unnatural to me to restrict to pseudomonoidal G. Anyway, for stating a monoidal yoneda's lemma it turns out to be an unnecessary assumption. $\endgroup$ Apr 27 at 12:44
  • $\begingroup$ Yes of course, but I mean in the question it is stated (OP only says "monoidal" but I suspect they mean strong monoidal) $\endgroup$ Apr 27 at 12:45
  • $\begingroup$ Yes, by "monoidal" I had meant "strong monoidal". Is this the same as "pseudomonoidal"? I've never come across that term before. $\endgroup$ Apr 27 at 14:24
  • $\begingroup$ Also, is there a missing condition in the statement of monoidal Yoneda? I would normally require that a monoidal natural transformation is compatible with binary tensor products and with the tensor unit, and this statement only seems to deal with binary tensor products. I would have expected that in order for a natural transformation to be monoidal, you also need the composite $I \to Gy \to G(I_A)$ to be equal to the map $I\to G(I_A)$ coming from lax monoidality of $G$, where $I_A$ is the tensor unit in $A$. Or something like this. $\endgroup$ Apr 27 at 14:31

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