Your organization currently has an onpremises mail system and you want to integrate that environment with ExchangeOnline.
Coexistence can either be cross-premises, supporting basic features such as a shared global address list, or hybrid, where there is full interoperability between the online and on-premises environments, including onboarding and offboarding facilities, free/busy integration, and delegation. Hybrid coexistence requires Directory Synchronization and typically would use single sign-on (SSO) for authentication.