Cofree coalgebra ($R(X)=(GX,{\delta }_X)$)

For a comonad, the cofree coalgebra on $X$ is $(GX,{\delta }_X)$. A map $UA\to X$ transposes to the coalgebra morphism $Gfa:A\to GX$. This gives the adjunction $U⊣R$ with the forgetful functor.