We define ordered sequential consistency (OSC), a correctness criterion for concurrent objects, which captures the typical behavior of many real-world services, e.g., ZooKeeper, etcd, Chubby, Doozer, and Consul. A straightforward composition of OSC objects is not necessarily OSC. To remedy this, we recently implemented a composition framework that injects dummy updates in specific scenarios. We prove that injecting such updates, which we call here leading updates, enables correct OSC composition. We generalize OSC to define G-OSC, a generic criterion for concurrent objects, which encompasses a range of criteria, including sequential consistency and linearizability.