There’s an adjunction between commutative monoids and pointed sets. That gives the comonad we want!

Hmm, not a very clear explanation? True. Read this for all the details:

https://golem.ph.utexas.edu/category/2022/10/theres_an_adjunction_between_c.html

@johncarlosbaez I'll make a guess about what's going on here, then have a look at your blog post to see if I got it right.

I reckon the comonad is on commutative monoids, which means there's a right adjoint from commutative monoids to pointed sets. Surely that has to be a forgetful functor that points to the identity element.

@johncarlosbaez Left adjoint to that will be a free construction, I.e. words over the elements of the set, quotiented by permutations and adding/removing the “pointed at” element.

So the comonad takes a commutative monoid and gives us a new one whose elements are “bags” of the original elements