We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: 1.On marked simplicial sets (due to Lurie [31]), 2.On bisimplicial spaces (due to deBrito [12]), 3.On bisimplicial sets, 4.On marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal–Tierney and the straightening construction due to Lurie.