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A comonad on $\mathbf{Set}$ is a tuple $(F,\epsilon,\delta)$ where $F\colon\mathbf{Set}\to\mathbf{Set}$ is a functor, $\epsilon\colon F\to\mathsf{id}_\mathbf{Set}$ and $\delta\colon F\to F\circ F$ are … comonads There are many polynomial comonads on $\mathbf{Set}$; in fact, a result of Ahman and Uustalu says that they are in bijection with categories: every category can be identified with a polynomial comonad …

For any monoid $(M,e,*)$ in $\mathsf{Set}$ there is a corresponding comonad $y^M$ on $\mathsf{Set}$. … Here are three more polynomial comonads for any set $S$: Store comonad (mentioned above), the functor $F(y)= Sy^S$. Linear comonad, the functor $F(y)=Sy$, with projection and diagonal. …