$\def\Cat{\mathbf{Cat}}\def\Set{\mathbf{Set}}\def\A{\mathcal{A}}$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.



The convolution of comonads is a comonad
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](https://arxiv.org/abs/1201.2991)), 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'](https://arxiv.org/abs/1507.08710).) 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.


