When creating test cases after unit testing(done by developers) QA team starts working on component testing. For component testing level there will be test cases written by the QA's

When it comes to component integration, should the QA team write test cases for component integration and so on system level, system integration etc?

If the answer is yes, please provide a little example on how to do it.
Currently I'm working on a test automation project and started creating test cases. With that I got this issue. Im planning on doing API testing and End to End testing. Happy to consider new ideas too.