The convolution of comonads is a comonad

Question

$\def\Cat{\mathbf{Cat}}\def\Set{\mathbf{Set}}\def\A{\mathcal{A}}$I stumbled into the following statement:

Let $\Cat(\Set,\Set)_s$ be the category of small functors[¹] $F : \Set \to\Set$ and let $F,G$ be two comonads on this category; then, the Day convolution $F * G$ is itself a comonad.

I find this both interesting and difficult to prove: I can come up easily with a counit: $$ F * G \overset{\epsilon * \epsilon'}\Rightarrow 1 * 1 = 1 $$ but what about comultiplication? The only way to build a valid enough candidate for a comultiplication $\sigma_{F*G}$ was to consider a rather complicated cowedge $$i_{UV} : [U\times V, A]\times FU\times GV \to F*G(F*G(A))$$ and subsequently, the map induced between coends seems to be a natural transformation $$\sigma_{F*G} : F*G \Rightarrow (F*G)(F*G) $$ it seems however extremely painful to prove that this is coassociative and counital.

Do you know this result, and a conceptual reason for it to be true? Or, in the worst case, a counterexample?

[¹]: An endofunctor of $\Set$ is called small if it results as the left Kan extension of a functor $\A\to\Set$, where $\A\subseteq\Set$ is a small subcategory, along said inclusion. This restriction is needed in order for $\Cat(\Set,\Set)$ to (exist and to) form a locally small category.

Answer 1

This is an instance of a standard fact about duoidal categories.

Proposition. Let $(\mathcal{V},\ast,J,\circ,I,\ldots)$ be a duoidal category (see e.g. Section 4.2 of Street's lecture notes), and let $(G,\delta,\varepsilon)$ and $(H,\delta,\varepsilon)$ be $\circ$-comonoids in $\mathcal{V}$. Then $G \ast H$ is a $\circ$-comonoid with comultiplication $\require{AMScd}$ \begin{CD} G\ast H @>\delta\ast\delta>> (G \circ G)\ast (H \circ H) @>\gamma>> (G \ast H) \circ (G \ast H) \end{CD} and counit \begin{CD} G\ast H @>\varepsilon\ast\varepsilon>> I \ast I @>\mu>> I. \end{CD} (Here $\gamma$ and $\mu$ are further parts of the duoidal structure on $\mathcal{V}$.)

Proof. By the definition of a duoidal category, the functor $\ast \colon \mathcal{V} \times \mathcal{V} \longrightarrow \mathcal{V}$ is opmonoidal (a.k.a. oplax monoidal) with respect to the $\circ$-monoidal structures, and therefore preserves comonoids by a standard argument (which yields the above expressions of the comultiplication and counit).

Example. The category of accessible endofunctors of $\mathbf{Set}$ is (normal) duoidal with $\circ$ given by composition and $\ast$ given by Day convolution, and so on. (See Section 8.1 of Garner & López Franco's paper 'Commutativity'.) Since the $\circ$-comonoids in this duoidal category are precisely the accessible comonads on $\mathbf{Set}$, the above Proposition implies that the Day convolution of two accessible comonads on $\mathbf{Set}$ is a comonad on $\mathbf{Set}$ in a canonical way.

Answer 2

As Alexander Campbell explained above, the answer comes from duoidality of $\circ$ over $*$. I'll focus instead on intuition.

Ahman and Uustalu showed that polynomial comonads on $\mathbf{Set}$ can be identified with categories. All the examples of comonads on $\mathbf{Set}$ I know of that are not polynomial are listed in the answers to this mathoverflow post. I'll concentrate on intuition from polynomial comonads.

Theorem: Let $c,d$ be polynomial comonoids and let $\mathbf{C}$ and $\mathbf{D}$ be the associated categories. Then the Day convolution $c*d$ is again polynomial, and the associated category is the cartesian product of categories, $\mathbf{C}\times\mathbf{D}$.

This can be found in my book with Nelson Niu.