The article
https://ncatlab.org/nlab/show/category+of+G-sets
about the category of continuous G-sets previously suggested that the category could be characterized as a category of coalgebras for a comonad. Does this come from the comonad induced by the adjuntion mentioned there? I.e. is the forgetful functor from continuous G-sets to G-sets comonadic? And what is the description of its purported right adjoint?
Does anyone have a reference to a paper?