A sort of Day convolution without enrichment

Question

Some time ago I was trying to define a monoidal structure on a functor category $[\mathcal{C},\mathcal{D}]$ between two monoidal categories $\mathcal{C}$ and $\mathcal{D}$, such that the monoid objects are exactly the monoidal functors (and if $\mathcal{C}$ and $\mathcal{D}$ are symmetric, then $[\mathcal{C},\mathcal{D}]$ becomes symmetric and the commutative monoids are the symmetric monoidal functors).

I came up with the following: assume that $\mathcal{D}$ is monoidally cocomplete (cocomplete with right exact tensor product). Given $x\in \mathcal{X}$, define $\mathcal{F}(x)$ the category of factorizations of $x$, whose objects are triples $(a,b,\varphi: a\otimes b\to x)$ and morphisms are pairs $(f,g):(a,b,\varphi)\to (a',b',\varphi')$ with $f:a\to a'$ and $g: b\to b'$ making the obvious diagram commute. Then if $F,G: \mathcal{C}\to \mathcal{D}$, we define $$(F\otimes G)(x) = \operatorname{colim}_{(a,b,\phi)\in \mathcal{F}(x)}F(a)\otimes F(b).$$

I think I correctly proved that this defined a monoidal structure on the functor category, with unit object $I:x\mapsto colim_{I_{\mathcal{C}}\to x} I_{\mathcal{D}}$, and that the monoid objects were indeed the monoidal functors (maybe I used some stronger hypotheses without realizing but I think it's fine).

Then I learned about Day convolution, which achieves the same goal, except that its definition seems to require that $\mathcal{C}$ is $\mathcal{D}$-enriched. I can't figure out whether those definitions are closely related (they seem to be, but I can't quite convince myself), and in particular whether they coincide when they are both defined.

Answer 1

Here is what I understand thanks to varkor's comment. (Disclaimer: I will not bother with associators, and will pretend that all monoidal operations are associative, as would any sane person.)

If $\mathcal{C}$ is a monoidal category, then given objects $x_1,\dots,x_n$ and $y$, you can form the "multi-hom" set $\operatorname{Hom}(x_1,\dots,x_n; y) = \operatorname{Hom}(x_1\otimes\cdots\otimes x_n, y)$. This sort of structure is abstracted away in the notion of multicategory, and then given some multicategory $\mathcal{C}$, you can ask whether it is representable, in other words whether the multimorphisms do come from a monoidal product as just shown.

It turns out that given any monoidal categories $\mathcal{C}$ and $\mathcal{D}$, there is a canonical multicategory structure on the functor category $[\mathcal{C}, \mathcal{D}]$. If $F_1,\dots,F_n$ and $G$ are functors $\mathcal{C}\to \mathcal{D}$, a multimorphism from $(F_1,\dots,F_n)$ to $G$ is simply a natural transformation between the two functors $\mathcal{C}^n\to \mathcal{D}$ defined by $$(x_1,\dots,x_n)\mapsto F_1(x_1)\otimes\cdots\otimes F_n(x_n)$$ and $$(x_1,\dots,x_n)\mapsto G(x_1\otimes\cdots\otimes x_n)$$ respectively (with obvious actions on morphisms).

Then we can say that there is a Day convolution product in $[\mathcal{C}, \mathcal{D}]$ if this multicategory is representable, and what I had shown was essentially that this is the case when $\mathcal{D}$ is monoidally cocomplete, with the monoidal product defined as in my question. In particular, by unsual uniqueness of representability, this coincides with the usual Day convolution when that makes sense (for instance when $\mathcal{D}$ is just the category of sets).

I will quickly sketch the proof to convince readers as well as myself. Let $F_1,F_2,G: \mathcal{C}\to \mathcal{D}$ be functors. A natural transformation $\theta: F_1\otimes F_2\Rightarrow G$ (with $F_1\otimes F_2$ defined as in my question) corresponds to morphisms $$\left(\operatorname{colim}_{(a,b,\varphi: a\otimes b\to x)}F_1(a)\otimes F_2(b)\right) \to G(x)$$ for all $x\in \mathcal{C}$, which means that for all morphisms of the form $\varphi: a\otimes b\to x$ in $\mathcal{C}$ we should define some $\theta_\varphi: F_1(a)\otimes F_2(b)\to G(x)$, with a bunch of somewhat obvious commutative diagrams stating that everything is natural. Taking $\varphi$ to be the identity of $a\otimes b$ yields morphisms $F_1(a)\otimes F_2(b)\to G(a\otimes b)$ which are easily shown to define a natural transformation from $(a,b)\mapsto F_1(a)\otimes F_2(b)$ to $(a,b)\mapsto G(a\otimes b)$. Conversely, given such a natural transformation, we can recover $\theta$ by taking $\theta_\varphi$ to be the composition of $F_1(a)\otimes F_2(b)\to G(a\otimes b)$ and the morphism $G(\varphi): G(a\otimes b)\to G(x)$.