Contracts vs. preferences over colleagues in matching

I introduce a general many-to-one matching framework which includes the matching with contracts model as well as models of matching with preferences over colleagues as special cases. I show that this general model can be embedded into the model with contracts, and the model with contracts can be embedded in the model with preferences over colleagues, thus the models are equivalent, and all results from the many-to-one matching with preferences over colleagues literature and the model with contracts literature can be applied to each other.