Finitary monads on $\operatorname{Set}$ are substitution monoids. Finitary monads on $\operatorname{Set}_*$ are...?

Question

$\DeclareMathOperator\Fin{Fin}\DeclareMathOperator\Lan{Lan}\DeclareMathOperator\Set{Set}$

The present question is intimately related to another question.

It is well known that the category of functors $\Fin \to \Set$ is equivalent to the category of finitary endofunctors $\Set \to \Set$; in this equivalence, finitary monads correspond to what are called substitution monoids on $[\Fin,\Set]$, i.e., to monoids with respect to the monoidal structure $$ F \diamond G = m\mapsto \int^n Fn \times G^{*n}m \tag{$\star$} $$ where $G^{*n}$ is the functor $$ m \mapsto \int^{p_1,\dotsc, p_n} Gp_1 \times \dotsb \times Gp_n \times \Fin(\sum p_i, m). $$ More precisely, the equivalence $[\Set,\Set]_{\omega} \cong [\Fin,\Set]$ can be promoted to a monoidal equivalence, and composition of endofunctors corresponds to substitution of presheaves in the following sense: let $J : \Fin \to \Set$ be the inclusion functor, then $$ \Lan_J(F\diamond G) \cong \Lan_JF \circ \Lan_JG \tag{$\heartsuit$} $$ and $$ (S\circ T) J \cong SJ \diamond TJ\tag{$\clubsuit$} $$ for two finitary endofunctors $S,T : \Set \to \Set$. (Kan extending along $J$ and precomposing an endofunctor of $\Set$ with $J$ is what defines the equivalence.)

I would like to prove the same exact theorem, replacing everywhere the cartesian category of sets with the monoidal category of pointed sets and smash product, but I keep failing.

The equivalence of categories $$ [\Fin_*,\Set_*]\cong [\Set_*,\Set_*] $$ remains true; and this equivalence must induce an equivalence between the category of finitary monads on pointed sets, and the category of suitable "pointed substitution" monoids, that are obtained from the iterated convolution on $[\Fin_*,\Set_*]$ as $$ F\diamond' G = m \mapsto \int^n Fn \land G^{*n}m $$ where $\land$ is the smash product, and $G^{*n}$ iterates the convolution on $[\Fin_*, \Set_*]$ induced by coproduct on domain, and smash on codomain: $$ G^{*n}m = \int^{p_1,\dotsc,p_n} Gp_1 \land \dotsb\land Gp_n \land \Fin_*(\bigvee p_i, m) $$ where $\bigvee p_i$ is the coproduct of pointed sets, joining all sets along their basepoint.

This would be the perfect equivalent of $(\star)$.

However, trying to prove the isomorphisms $(\heartsuit)$, $(\clubsuit)$, I find that it is not true that $\Lan_J(F\diamond G) \cong \Lan_JF \circ \Lan_JG$. I am starting to suspect that the generalisation is false as I have stated it, or that it is true in a more fine-tuned sense.

So, I kindly ask for your help:

To what kind of monoids on $[\Fin_*,\Set_*]$ do finitary monads on pointed sets correspond?

Edit: I am led to believe this construction is not a particular instance of an enriched Lawvere theory, because in that framework a theory isn't what it is in mine:

• for Power, if $\mathcal V$ is a locally finitely presentable base of enrichment, a theory is an identity on objects functor $\mathcal V_\omega ^{op}\to \mathcal L$ from the subcategory of finitely presentable objects to $\mathcal L$ strictly preserving cotensors; if $\mathcal V = \Set_*$ with smash product, cotensors in $\mathcal V^{op}$ are tensors in $\mathcal V$, thus smash products.
• Instead, for me, a theory is a bijective-on-objects functor $\Fin_* \to \mathcal L$ that sends coproduct into smash product (or even a more general monoidal structure on $\mathcal L$).

Or at least, this is what I was led to believe trying to back-engineer the equivalence between finitary monads and Lawvere theories in the case of pointed sets.

Answer 1

Rather than checking to see if that is a substitution monoid, I think you might have an easier time using Rory Lucyshyn-Wright’s notion of a eleutheric system of arities, seen here. It’s a relatively straightforward condition to check, with several equivalent statements.

Edit: To give a (very brief) description of how this works out: a monoidal subcategory of $\mathcal{V}$ is a symmetric monoidal subcategory $j:\mathcal{J} \to \mathcal{V}$, and a $\mathcal{J}$-ary Lawvere theory is a $\mathcal{V}$-category with a bijective-on-objects, power-preserving functor $\mathcal{J} \to \mathcal{T}$. A system of arities is eleutheric if for every $T: \mathcal{J} \to \mathcal{V}$, the left Kan extension $Lan_jT$ exists and is preserved by $\mathcal{V}(J,-), J \in \mathcal{J}$. This ends up being just enough to ensure that every $\mathcal{J}$-ary theory induces a free $\mathcal{T}$-algebra monad on $\mathcal{V}$.