= Category of coalgebras for a comonad
{title2=$\mathcal E^G$}
The category $\mathcal E^G$ consists of <coalgebras for a comonad> and their structure-preserving morphisms. Its <forgetful functor> has the <cofree coalgebra> as right adjoint. If $\mathcal E$ is a <topos> and $G$ preserves <finite limits>, those limits are created by the forgetful functor, while <exponentials in a coalgebra topos> and the <subobject classifier of a coalgebra topos> can be constructed as equalizers inside cofree objects.
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