An example of an equivalence which is not an adjunction

Question

In this question it is stated that an equivalence is not necessarily an adjunction, since the natural isomorphism need not satisfy the triangle identities, and may need to be "improved" to give an adjunction. What is a natural example of an equivalence which is not an adjunction.

Answer 1

I do not know any "natural example", but here are some examples:

Consider two monoids $M,N$ as one-object categories. An equivalence of categories between $M$ and $N$ is a pair of homomorphisms $f : M \to N$, $g : N \to M$ together with invertible elements $\alpha \in M^{\times}$, $\beta \in N^{\times}$ such that $g(f(m))=\alpha m \alpha^{-1}$ for all $m \in M$ and $f(g(n))=\beta n \beta^{-1}$ for all $n \in N$. In fact, $\alpha$ and $\beta$ correspond to isomorphisms of functors $\alpha : \mathrm{id}_M \to g \circ f$ and $\beta : \mathrm{id}_N \to f \circ g$. The first triangle identity is $f \bullet \alpha = \beta \bullet f$, which comes down to $f(\alpha)=\beta$. The second becomes $g(\beta)=\alpha$. Now consider the example $M=N$, $f = g = \mathrm{id}_M$. It follows that an equivalence of this type consists of two central invertible elements $\alpha$, $\beta$, and it is an adjoint equivalence iff $\alpha=\beta$. So for example any non-trivial abelian group gives an example of an equivalence which is not an adjoint equivalence.