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Ahman et al. wrote about when a container is a comonad. Containers can also be monads, such as List. This means that we can take all containers that are endofunctors on Set and they live in the endofunctor category on set. This category has a monoidal product, which is functor composition. Thus we can have a monoidal category for containers that are either monads or comonads. This must have a diagrammatic calculus. What are the axioms of this diagrammatic calculus?

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    $\begingroup$ Don't you need a distributive law to compose monads? $\endgroup$
    – Gerrit Begher
    Commented Feb 2, 2018 at 0:28
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    $\begingroup$ Not every container is a monad; a container is just a polynomial functor, it may or may not have a monad structure. The category of polynomial functors is indeed a monoidal category under functor composition, so its calculus is just that of an ordinary (non-symmetric) monoidal category. (Polynomial) monads and comonads are just monoids and comonoids in this monoidal category. $\endgroup$
    – Mike Shulman
    Commented Feb 2, 2018 at 17:20

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