Does anyone know anything about when functors E:C-->D have the property that the functor Lan_E:[C,Set]-->[D,Set] given by left Kan extension is comonadic? Actually I'm interested in something a bit more general. Let E:C-->D be a functor. I'm happy to suppose that it is bijective on objects and faithful. Then Lan_E has a right adjoint, given by restriction along E. Let W be the induced comonad on [D,Set], and [D,Set]^W its category of coalgebras. The comparison K:[C,Set]-->[D,Set]^W has a right adjoint R, constructed using equalizers in [C,Set]. Comonadicity would mean that this adjunction is an equivalence; what I really want to know is when/whether R is fully faithful, so that K is a reflection onto a full subcategory. (This is equivalent to Lan_E preserving the equalizers which are used to construct R.) This seems like something topos theorists might know about. Steve.