$\def\Fin{\text{Fin}\_\*}
\def\Set{\text{Set}\_\*}
\def\dd{\mathop{\diamond\_\land}}$ 



> 
> The present question is intimately related to [another question](https://mathoverflow.net/questions/361690/finitary-monads-on-operatornameset-are-substitution-monoids-finitary-monad).
> 
> 
> 


Let $\Fin$ be the category of pointed sets.


The category $[\Fin,\Set]$ of $\Set$-enriched functors has a convolution monoidal product when both categories are endowed with the smash product operation.


This means that we can convolve two parallel functors $F,G : \Fin \to \Set$ to
$$
F\*G := n\mapsto \int^{pq}Fp\land Gq\land \Fin(p\land q,n)
$$
One would be tempted to define, starting from this, a substitution product following Kelly's "[On the operads of PJ May](http://www.tac.mta.ca/tac/reprints/articles/13/tr13abs.html)": the iterated convolution
$$
F^{\ast m} \cong \int^{p\_1,...,p\_m} Fp\_1\land \dots \land Fp\_m \land \Fin(p\_1\land\dots\land p\_m,\\_)
$$ should define a substitution product
$$
F \dd G := n\mapsto \int^m Fm\land G^{\ast m}n
$$
However, this is not an associative operation: in Kelly's proof, the fact that convolution and substitution interact in the following way is crucial:
$$
(F\dd G)^{\ast m} \cong F^{\ast m}\dd G\tag{$\star$}
$$
Because once $(\star)$ is proved, it follows that
$$
\begin{align\*}
 (F\dd G)\dd H & = \int^n (F\dd G)n\land H^{\*n} \\
 & \cong \int^{nm} Fm\land G^{\*m}n\land H^{\*n} \\
 F \dd (G \dd H) & = \int^m Fm \land (G\dd H)^{\*m} \\
 & \cong \int^m Fm \land (G^{\*m}\dd H) \\
 & \cong \int^{nm} Fm\land G^{\*m}n\land H^{\*n}.
\end{align\*}
$$


Let's try to prove $(\star)$ then, adopting a slightly improper notation in order to save space: $\underline p$ is a tuple of the appropriate length, and we write $\int^{\underline{p}} F\underline p \land \Fin(\underline p^\land,\\_)$ for the integral defining $F^{\ast m}$:
$$
 \begin{align\*}
 (F\diamond\_\land G)^{\ast m} & =\int^{\underline{k}} \Fin(\underline{k}^\land, \\_)\land (F\diamond\_\land G)(\underline{k}) \\
 & \cong \int^{\underline{k}, \underline{p}} \Fin(\underline{k}^\land, \\_) \land
 \left(Fp\_1 \land G^{\ast p\_1}k\_1\right) \land \cdots \land
 \left(Fp\_m \land G^{\ast p\_m}k\_m\right) \\
 & \cong \int^{\underline{p}} Fp\_1\land\cdots\land Fp\_m \land (G^{\ast p\_1} \ast\cdots\ast G^{\ast p\_m}) \\
 & \cong \int^{\underline{p}} Fp\_1\land\cdots\land Fp\_m \land G^{\ast \sum p\_i} \\
 & \cong \int^{\underline{p}} Fp\_1\land\cdots\land Fp\_m \land \int^r G^{\ast r}\land \Fin(\sum p\_i,r) 
 \end{align\*}
 $$
whre I expanded the definition, rearranged terms, compacted again the definition and finally used the fact that $(\\_)^{\ast \\_} : [\Fin,\Set]\times \Fin^\text{op} \to [\Fin,\Set]$ is a bifunctor, so that by ninja Yoneda $G^{\ast k} \cong \int^r G^{\ast r}\land \Fin(k,r)$.


Now I'm stuck, because $\Fin(\sum p\_i,r) \not\cong \Fin(\bigwedge p\_i,r) $; the monoidal structure given by coproduct is different from the one given by smash product; the first categorifies the monoid structure $(\mathbb N, +)$, whereas the second categorifies the monoid structure $(\mathbb N, \circ)$, where $p\circ q := pq-p-q+1$ (if $p,q$ are finite sets with $p,q$ elements; or counting in another way, if $p:= \{0,...,p\}$ then $p\circ q=pq$).


If I had obtained $\Fin(\bigwedge p\_i,r)$ instead of $\Fin(\sum p\_i,r)$, all would have been ok, because now
$$
 \begin{align\*}
 & \cong \int^{\underline{p}} Fp\_1\land\cdots\land Fp\_m \land \int^r G^{\ast r}\land \Fin\big(\bigwedge p\_m,r\big) \\
 & \cong \int^{\underline{p},r} \Fin(\underline{p}^\land,r) \land Fp\_1\land\cdots\land Fp\_m \land G^{\ast r} \\
 & \cong \int^r F^{\ast m} r \land G^{\ast r} \\
 & \cong F^{\ast m} \diamond\_\land G
 \end{align\*}
 $$


It seems then that there is no way to define an operad-like monoidal structure on $[\Fin,\Set]$, neither starting from the framework of [my previous question](https://mathoverflow.net/questions/361690/finitary-monads-on-operatornameset-are-substitution-monoids-finitary-monad), nor in a purely $\Set$-enriched one, i.e. taking the smash-product monoidal structure on both domain and codomain of $[\Fin,\Set]$.



> 
> Why? What went wrong?
> 
> 
> 



Substitution structure on pointed sets
In the same vein as my response to your other question, if pointed finite sets are an eleutheric system of arities, Lawvere theories over that system of arities will be equivalent to monads in a certain monoidal category. This is in section 11 of Lucyshyn-Wright [here](https://arxiv.org/pdf/1511.02920.pdf).


Edit: I briefly outlined how the relationship between eleutheric systems of arities and these monads works [here](https://mathoverflow.net/a/361975/75783). I haven't spent a huge amount of time working out this part of the paper, so I don't want to risk giving a bad answer. Sections 9,10,11 of Lucyshyn-wright's paper give a very detailed construction of this correspondence.


