Cartesian comonad ($(G,\epsilon ,\delta )$)

In the finite-limit setting, a Cartesian comonad is a comonad whose underlying endofunctor preserves finite limits. Its category of coalgebras for a comonad has finite limits created by the forgetful functor. On an elementary topos, equalizer constructions of exponentials in a coalgebra topos and the subobject classifier of a coalgebra topos make that coalgebra category an elementary topos.