TITLE: How the category of continuous G-sets can be seen as the comonad algebra?
QUESTION [6 upvotes]: 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?

REPLY [2 votes]: This is just a combination of three basic results. For one, the nLab entry on the category of $G$-sets shows that the adjoint pair $i_G\dashv r_G$ is a geometric morphism between toposes, where $i_G$ is the inclusion of $G$-sets into $G_{disc}$-sets and where $r_G$ is its right-adjoint. Now, a geometric morphism is surjective if and only if it is comonadic if and only if the left adjoint $i_G$ is faithful, which is easy to see (and also follows from the stuff proven on the nLab page).
Thus, $(G\text{-Set})$ is naturally equivalent to the category of coalgebras for the comonad $i_G\circ r_G\colon(G_{disc}\text{-Set})\to(G_{disc}\text{-Set})$.