Comonadic base change for enriched categories

For our concepts of change of base and comonadicity, we work in the general context of the tricategory Caten whose objects are bicategories V and whose morphisms are categories enriched on two sides. For example, for any monoidal comonad G on a cocomplete closed monoidal category C, the forgetful functor U:C^G→C is comonadic when regarded as a morphism in Caten between one-object bicategories. Other examples are provided including that obtained from any comonoidal C-enriched category. We show that the forgetful pseudofunctor U:V^G→V from the bicategory of Eilenberg-Moore coalgebras for a comonad G on V in Caten induces a change of base pseudofunctor U˜:V^G-Mod→V-Mod which is comonadic in a bigger version of Caten. We should emphasise that the right adjoints to U and U˜ generally do not have right adjoint lax functors, confirming our need to work with two-sided enrichments. We define Hopfness for such a comonad G and prove that having that property implies U creates left (Kan) extensions in the bicategory V^G. We provide conditions under which Hopfness carries over from G to the comonad G˜=U˜∘R˜ generated by the adjunction U˜⊣R˜. This has implications for characterizing the absolute colimit completion of V^G-categories. A motivating example was the monoidal category of differential graded abelian groups obtained as the category of coalgebras for a Hopf monoid in the category of abelian groups. Examples include some involving base bicategories V=Spn(E) of spans in an ordinary category E with pullbacks.