Coalgebras and modal logic for parametrised endofunctors

We study categories of coalgebras for endofunctors, which additionally depend on a parameter category. The corresponding category of coalgebras then naturally appears as cofibred over the parameters. We give examples of constructions in the cofibred framework and study the overall structure of such cofibrations. Moreover, the dependency of (modal) logics for coalgebras on a parameter category is investigated and shown to give rise to the dual of an institution.