I know that whether something is obvious or not is highly subjective and depends on the perspective. Nonetheless, some things are collectively regarded to be less obvious than others.
I also know that left adjoints are unique up to a unique isomorphism. An isomorphism may not be obvious, however. Some less immediate examples from basic algebra are listed in https://math.stackexchange.com/a/1268003/229174 and some examples of isomorphic categories are listed in https://mathoverflow.net/questions/13995/nontrivial-isomorphisms-of-categories.
When talking about natural isomorphisms in a functor category, however, I don't know of an example where the isomorphism would need a lot of thought. For example, a free abelian group (left adjoint to $U: \mathbf{Ab} \to \mathbf{Set}$) can be defined either directly as a set of finite multisets or as the abelianization (left adjoint to $U: \mathbf{Ab} \to \mathbf{Grp}$) of the free group (left adjoint to $U: \mathbf{Grp} \to \mathbf{Set}$). The proof of their equivalence is both simple and intuitive, however.
I'm looking for counterintuitive examples to natural isomorphisms, ideally for left adjoints.
