We propose a coalgebraic definition of weak bisimulation for a class of coalgebras obtained from bifunctors in~$SetCat$. Weak bisimilarity for a system is obtained as strong bisimilarity of a transformed system. The transformation consists of two steps: First, the behaviour on actions is expanded to behaviour on finite words. Second, the behaviour on finite words is taken modulo the hiding of invisible actions, yielding behaviour on equivalence classes of words closed under silent steps. The coalgebraic definition is justified by two correspondence results, one for the classical notion of weak bisimulation of Milner and another for the notion of weak bisimulation for generative probabilistic transition systems proposed by Baier and Hermanns.