 Comonad

A comonad is an endofunctor $G$ with natural transformations $ϵ:G\to 1$ and $\delta :G\to G^2$ satisfying $Gϵ\delta ={ϵ}_G\delta =1_G$ and $G\delta \delta ={\delta }_G\delta$. It is the categorical dual of a monad. Its coalgebras for a comonad carry compatible maps into their image under $G$.