Examples of non-adjoint equivalences 
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.
 A: Let's start with a functor $U:{\mathcal A}\to{\mathcal X}$ that's full and faithful. Bypassing questions of Choice, it's also essentially surjective on objects in the sense that there is a function $F:{\mathsf ob}{\mathcal X}\to{\mathsf ob}{\mathcal A}$ along with an assignment of an isomporhism $\eta_X:X\to U F X$ to each object $X$.
From these data, for each morphism $f:X\to Y$ of $\mathcal X$, we define $F f:F X\to F Y$ as the unique $\mathcal A$-map such that
$$ \eta_X^{-1};f;\eta_Y = U(F f), $$
in other words such that $\eta$ is natural.   It is easy to check that $F$ preserves identity and composition.
It remains to define $\epsilon$.  For one of the triangle laws we require $U\epsilon_A=\eta_{U A}^{-1}$, which uniquely defines $\epsilon_A$ since $U$ is full and faithful. Naturality of $\epsilon$ follows from that of $\eta$.
The other triangle law is $\eta_{U A};U\epsilon_A={\mathsf id}$, for which it suffices that this hold with $U$ applied, since that's full and faithful.
By naturality of $\eta$ and the first triangle law, we have
$$ \eta_X;U F\eta_X; U\epsilon_{F X} = \eta_X;\eta_{U F X}; U\epsilon_{F X} = \eta_X $$
but $\eta_X$ is invertible an $U$ is full and faithful, so the other triangle law holds.
In other words, the obvious data for "full, faithful and essentially surjective on objects" yield an adjoint equivalence.
So what other kind of equivalence is there? If the isomorphism $\eta$ in "essential surjectivity" is to be natural, it can only be as above.  However, the other isomorphism $\epsilon'$ could come from somewhere else.  Nevertheless, $\eta_{U A};U\epsilon'_A$ is still a natural automorphism of $U A$, which must be $U$ applied to a natural isomorphism of $A$.
In other words, a non-adjoint equivalence is given by arbitrary natural isomorphisms applied to an adjoint equivalence.
This is a situation that can easily be realised with group isomorphisms, yielding the counterexample that was requested.
