TITLE: Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?
QUESTION [15 upvotes]: It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid in $[\mathcal C, \mathcal V]$ is the same thing as a lax monoidal functor $\mathcal C \to \mathcal V$, where $[\mathcal C,\mathcal V]$ carries the monoidal structure given by the Day convolution.
Is the following also true?

If $\mathcal C,\mathcal D$ are monoidal $\mathcal V$-enriched categories, where $\mathcal V$ is cocomplete, then a lax monoidal functor $\mathcal C \to [\mathcal D,\mathcal V]$ is the same thing as a lax monoidal functor $\mathcal C \otimes \mathcal D \to \mathcal V$.

(In particular, if $\mathcal C$ is the unit category, then we get the original formulation.)
It certainly seems to be the case: if $\mathcal F \colon \mathcal C \otimes \mathcal D \to \mathcal V$ is a lax monoidal functor, then we have natural coherences
\begin{gather}
\mathcal F(c,\_)\otimes_{\text{Day}}\mathcal F(c',\_) \to \mathcal F(c\otimes c',\_) \\
I_{\text{Day}} \to F(I, \_)\,,
\end{gather}
where the multiplicative coherence is given by the composite
\begin{align}
& \int^{d,d'\colon\mathcal D}(\mathcal F(c,d)\otimes\mathcal F(c',d'))\otimes \mathcal D(d\otimes d',x)\\
\to & \int^{d,d'\colon\mathcal D}\mathcal F(c\otimes c',d\otimes d')\otimes\mathcal D(d\otimes d',x)\\
\to & \mathcal F(c\otimes c',x)\,.
\end{align}
and the unital coherence $\mathcal D(I_{\mathcal D},x)\to\mathcal F(I_{\mathcal C},x)$ comes (via the enriched Yoneda lemma) from the monoidal unit $I_{\mathcal V}\to\mathcal F(I_{\mathcal C},I_{\mathcal D})$ for $\mathcal F$.
I have not checked whether these satisfy the coherence conditions for a monoidal functor, but I would be surprised if they did not.
In the other direction, if $\mathcal F\colon \mathcal C \to [\mathcal D,\mathcal V]$ is lax monoidal, then we have coherences given by
\begin{align}
&\mathcal G(c)(d) \otimes \mathcal G(c')(d')\\
\cong & \int^{e,e'\colon\mathcal D} (\mathcal G(c)(e) \otimes \mathcal G(c')(e'))
\otimes (\mathcal D(e,d) \otimes \mathcal D(e',d')) \\
\to & \int^{e,e'\colon\mathcal D} (\mathcal G(c)(e) \otimes \mathcal G(c')(e'))
\otimes \mathcal D(e\otimes e',d\otimes d')\\
= & (\mathcal G(c) \otimes_{\text{Day}}\mathcal G(c'))(d \otimes d')\\
\to &\mathcal G(c\otimes c')(d\otimes d')
\end{align}
and monoidal unit $I_{\mathcal V}\to \mathcal F(I_{\mathcal C},I_{\mathcal D})$ by the enriched Yoneda lemma as before.
I haven't checked whether these coherences actually satisfy the appropriate diagrams, nor whether these two maps are indeed inverses.  
Is this fact true?  And if so, has it been proved in the literature somewhere?

REPLY [6 votes]: Said slightly differently, this is the observation that monoidal profunctors $J\colon A \to B$, that is lax monoidal functors $J\colon A^{op} \times B \to V$, correspond (up to isomorphism) to lax monoidal functors $j\colon B \to \hat A$, where $\hat A = [A^{op}, V]$ is equipped with Day-convolution. Each corresponding pair $J$ and $j$ comes with an isomorphism of monoidal profunctors
$$\hat A(ya, jb) \simeq J(a, b),$$
where $y\colon A \to \hat A$ is the yoneda embedding (which is pseudomonoidal wrt. Day-convolution).
This you can think of as a "monoidal Yoneda's lemma" and, formally, as being part of some kind of yoneda structure. This is the main motivation for https://arxiv.org/abs/1511.04070.
To formalise the above you need a setting allowing two types of morphism (mon. profunctors and lax mon. functors). This is why I used a form of generalised double categories $K$, instead of 2-categories which are traditionally used for yoneda structures.
One of main results states that for a "reasonable" double monad $T$ on $K$, given a yoneda embedding $y\colon A \to \hat A$ in $K$, a $T$-algebra structure on $A$ induces such a structure on the presheaf object $\hat A$, making $y$ into a yoneda embedding in the "double category" $Alg(T)$ of $T$-algebras. Taking $T =$ "free monoidal $V$-cats"-monad then recovers the correspondence above.
You can also apply this result to get yoneda embeddings for double categories and for categories $A\colon S^{op} \to Cat$ indexed over some small category $S$. In both cases you get presheaf objects that are simpler than the classical ones (given by resp. Bob Pare (http://www.tac.mta.ca/tac/volumes/25/17/25-17.pdf) and e.g. Mark Weber (https://link.springer.com/article/10.1007/s10485-007-9079-2) )
You should also be able to get/recover yoneda embeddings for "cocomplete modular" closure/topological/approach spaces (in the sense of https://arxiv.org/abs/1704.00209) by taking $T$ to be some extension of the power-set or ultrafilter monad, but I haven't had the time to work that out.
Another main result proves that the "lifted" algebraic yoneda embedding defines $\hat A$ as the free cocompletion of $A$ in $Alg(T)$. In the monoidal case this recovers a result of Kelly and Im given in https://www.sciencedirect.com/science/article/pii/0022404986900058.
I hope this helps. It has been good for me to find references to similar results here.