Congratulations on your first Pull Request and welcome to the Apache Airflow community! If you have any issues or are unsure about any anything please check our Contribution Guide (https://github.com/apache/airflow/blob/master/CONTRIBUTING.rst)
Here are some useful points:

• Pay attention to the quality of your code (flake8, pylint and type annotations). Our pre-commits will help you with that.
• In case of a new feature add useful documentation (in docstrings or in docs/ directory). Adding a new operator? Check this short guide Consider adding an example DAG that shows how users should use it.
• Consider using Breeze environment for testing locally, it’s a heavy docker but it ships with a working Airflow and a lot of integrations.
• Be patient and persistent. It might take some time to get a review or get the final approval from Committers.
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Docker Compose supports condition field. This prevents the container from running until the service reports that it is healthy.
https://github.com/docker/compose/blob/c81046aac0ce1360f1bab64368f9a318c593d82a/compose/config/compose_spec.json#L184-L195
https://github.com/docker/compose/blob/0bea52b18dda3de8c28fcfb0c80cc08b8950645e/compose/service.py#L756-L757
It is also supported by compose spec:
https://github.com/compose-spec/compose-spec/blob/b7fb5b65fff19651ecdb2d4d3232b01fb6866f70/schema/compose-spec.json#L184-L196
To me, it looks like an issue in the docker documentation.

See also: #13446

Thanks for correcting me here. I'll see if there are some tests or something over on the Docker side and attempt to clarify that documentation. I made a quick little test locally, and the "depender" service timed out while waiting for the "dependee" service to become healthy.