What are some examples of equivalences whose canonical unit/counit fail to satisfy the triangle identities?

It is common knowledge that not all equivalences satisfy the triangle identities, but that any equivalence can be refined by swapping out its unit (or counit) with a different one to form an adjoint equivalence which does satisfy the triangles while leaving both functors intact, so all functors that are part of an equivalence are also part of an adjoint equivalence.

I'm curious about equivalences where the canonical unit and counit do not satisfy the triangle identities -- the meaning of canonical here is hopefully canonical, but to be more precise I mean that the unit and counit that are 'obvious' to write down do not satisfy the triangles and need to be modified using the refinement to adjoint equivalences to do so.