I may be wrong, but we should be able to write the Bag monad in a polynomial form. The bag monad, is exectly the multiset monad whose category of algebras are the commutative monoids. Another name for the bag monad is a container.

Containers are synonymous with polynomail functors. The data that defines a container are precisely the following morphisms in a locally Cartesian closed category $C$:

$$ 1 \xleftarrow{\text{f}} X \xrightarrow{\text{g}} Y \xrightarrow{\text{h}} 1 $$

Where $1$ is the terminal object in $C$. This defines an endofunctor for which there is a monad. Specifically, the endofunctor is :

$$ C/W \xrightarrow{\text{f^* }} C/X \xrightarrow{\Pi_g} C/Y \xrightarrow{\Sigma_h} C/Z $$

We are interested in endofunctors so $W$ and $Z$ are $1$ in $Set$.

What is the polynomial form of the bag monad?