Your non-example of vector spaces and their duals can be souped up to a real example.

Let $C$ be the groupoid of finite-dimensional vector spaces and linear isomorphisms. Then there are two obvious functors $C \to C^{op}$: the linear dual, and the natural isomorphism $C \stackrel \sim \to C^{op}$ that one has for any groupoid. These functors are unnaturally isomorphic.