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The category of polynomial functors on Set is equivalent to the category of containers.

We have a prescription for when a container is a comonad.

There are a few other questions that come to mind. The first question in my mind, though, was, given you have a polynomial functor on Set, what properties ensure it is a monad?

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  • $\begingroup$ The title of the paper at the second link is catchy, but it seems a bit of a linguistic abuse to characterize its contents as describing when a container "is" a comonad. Since a single container can support multiple comonad structures (cf examples 3.2 and 3.3). Perhaps a clearer informal description of the contents would be "a characterization of a comonad whose underlying endofunctor is the interpretation of a container in terms of a structure that 'feels' more natural in the context of containers" $\endgroup$
    – Gabriel C. Drummond-Cole
    Commented Jan 10 at 3:08
  • $\begingroup$ If you agree with this characterization, then perhaps your question could be reframed as something like "what is a characterization of a monad whose underlying endofunctor is a polynomial functor on Set in terms of a structure that 'feels' more natural in the context of polynomial functors?" $\endgroup$
    – Gabriel C. Drummond-Cole
    Commented Jan 10 at 3:10

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