Yoneda Lemma for monoidal functors 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.
 A: 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.
