I always see unit tests as a snapshot of dynamic behavior by 'recording' the tested logic. I make sure I'm aware of any logic changes since that change could eventually break the desired result.

Whenever I'm vetting in-/output values to a given set of parameters, I do move from white box (unit tests) to grey box testing.

When I'm done with grey/white box tests, I do make sure integration works as expected.

Why all the hustle of moving through 'the onion'? I wanna make sure to detect misbehaving/unexpected logic as quickly as possible. Searching for malfunction detected while running integration tests takes way more time than already catching them at the onion's most inner layer (unit tests).