Complex sentence: The contracts were "dual" in that they were signed between the City and two separate private companies (the Interborough Rapid Transit Company and the Brooklyn Rapid Transit Company), all working together to make the construction of the Dual Contracts possible.
Split into simpler sentences: The contracts were "dual" in that they were signed between the City and two separate private companies. Both the IRT and BRT (later Brooklyn--Manhattan Transit Corporation, or BMT) worked together to make the construction of the Dual Contracts possible.