Institutions with proof-theoretic structure, here called 'institutions with proofs', provide a complete formal notion for the intuitive notion of logic, including both the model and the proof theoretic sides. This paper introduces a concept of proof rules for institutions and argues that the proof systems of the actual institutions with proofs are freely generated by their presentations as systems of proof rules. We also show that proof-theoretic quantification, an institutional refinement of the (meta-)rule of Generalization from classical logic, can also be added freely to any proof system. By applying these universal properties, we are able to provide some general compactness results for proof systems and some general soundness results for institutions with proofs. We also discuss several open problems and further research directions.