Pushouts of categories, derived limits and colimits

In a paper by Ford, it is claimed that to any pushout square of categories with all involved functors injective, there is associated an exact “Mayer–Vietoris” sequence of derived (co)limits. We provide a counter-example to this general statement. Further, we construct the Mayer–Vietoris sequence under some restrictions that cover the well-known case of a pushout square of group monomorphisms.