It is well known (e.g., [Reference for "lax monoidal functors" = "monoids under Day convolution"](https://mathoverflow.net/questions/130616/reference-for-lax-monoidal-functors-monoids-under-day-convolution/130619) ) 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?



Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?
**Edit:** Alexander Campbell points out in the comments that a reference for this, in the case ${\cal V}=\rm Set$, is Claudio Pisani's paper [Sequential multicategories](http://tac.mta.ca/tac/volumes/29/19/29-19abs.html). Following is a sketch of the argument.


Consider the category $\mathcal{V}\text{-}\mathrm{Mult}$ of $\mathcal{V}$-enriched *multicategories*. This includes the category $\mathcal{V}\text{-}\mathrm{MonCat}\_l$ of monoidal $\mathcal{V}$-categories and lax monoidal functors as a full subcategory, where the underlying multicategory of $\mathcal{C}$ has multimorphisms defined by $U\mathcal{C}(c\_1,\dots,c\_n; x) = \mathcal{C}(c\_1\otimes\cdots\otimes c\_n, x)$.


Furthermore, I claim the category $\mathcal{V}\text{-}\mathrm{Mult}$ has a "pointwise" monoidal structure, which $\mathcal{V}\text{-}\mathrm{MonCat}\_l$ is closed under. Given $\mathcal{V}$-multicategories ${\cal C,D}$, define the objects of ${\cal C\otimes D}$ to be pairs $(c,d)$ of objects of $\cal C$ and $\cal D$ respectively, with


$$({\cal C\otimes D})((c\_1,d\_1),\dots,(c\_n,d\_n);(x,y)) = {\cal C}(c\_1,\dots,c\_n;x) \otimes {\cal D}(d\_1,\dots,d\_n;y).$$


Note that this is *not* the "Boardman-Vogt tensor product" of multicategories/operads; in particular the latter only makes sense for symmetric multicategories, but this one works even for non-symmetric ones.


Now your claim can be re-expressed as


$$\mathcal{V}\text{-}\mathrm{Mult}({\cal C}, [{\cal D,V}]) \cong \mathcal{V}\text{-}\mathrm{Mult}({\cal C\otimes D,V}),$$


i.e. that the underlying multicategory of the Day convolution monoidal category $[{\cal D,V}]$ is an internal-hom for this tensor product on $\mathcal{V}\text{-}\mathrm{Mult}$. This should be straightforward because both of these multicategories have a direct description.


Of course the objects of the Day convolution monoidal category are the functors between the underlying *ordinary* categories $\cal D,V$. You might expect at first that the objects of the internal-hom in ${\cal V}\text{-}\mathrm{Mult}$ should instead be multicategory functors $\cal D\to V$, but in fact this is not true, because the "unit multicategory" is not the same as the "multicategory freely generated by an object", and it is maps out of the latter that detect the objects of the internal-hom. In the multicategory $\cal O$ freely generated by an object there are no $n$-ary morphisms for $n\neq 1$, and the unique object has only its identity unary morphism; thus a multicategory functor $\cal O \otimes D\to V$ acts only on the objects and unary morphisms of $\cal D$, and hence is also just a functor between underlying ordinary categories.


Now the $n$-ary morphisms in the internal-hom are the same as morphisms $\cal T\_n\otimes D \to V$, where $\cal T\_n$ is the "free-living $n$-ary morphism". Thus a morphism $(F\_1,\dots, F\_n) \to G$ consists of, for each $n$-ary morphism $(d\_1,\dots,d\_n) \to y$ in $\cal D$, an $n$-ary morphism $(F\_1(d\_1),\dots,F\_n(d\_n)) \to G(y)$ in $\cal V$, subject to naturality with respect to unary morphisms (because of the identity arrows in $\cal T\_n$). In the enriched case we have to rewrite that as a definition of a hom-object:


$$[{\cal D,V}](F\_1,\dots,F\_n;G) = \int\_{d\_1,\dots,d\_n,y} [{\cal D}(d\_1,\dots,d\_n;y),{\cal V}(F\_1(d\_1),\dots,F\_n(d\_n);G(y))].$$


On the other hand, the Day convolution tensor product is defined as


$$(F\_1\otimes F\_2)(y) = \int^{d\_1,d\_2} F\_1(d\_1)\otimes F\_2(d\_2) \otimes {\cal D}(d\_1\otimes d\_2,y)$$


so that


$$\begin{align}[{\cal D,V}](F\_1\otimes F\_2,G) &= \int\_{y} {\cal V}\left(\int^{d\_1,d\_2} F\_1(d\_1)\otimes F\_2(d\_2) \otimes {\cal D}(d\_1\otimes d\_2,y),G(y)\right)\\
&\cong \int\_{y,d\_1,d\_2} {\cal V}(F\_1(d\_1)\otimes F\_2(d\_2) \otimes {\cal D}(d\_1\otimes d\_2,y),G(y))\\
&\cong \int\_{y,d\_1,d\_2} [{\cal D}(d\_1\otimes d\_2,y),{\cal V}(F\_1(d\_1)\otimes F\_2(d\_2),G(y))]
\end{align}$$


which coincides exactly with the other version when $n=2$ and $\cal D,V$ are representable. A similar argument works for arbitrary $n$. So far we haven't used that $\cal D,V$ are monoidal rather than arbitrary multicategories, but apparently to construct composition one needs $\cal D$ to be at least promonoidal.


