ct.category theory - Is a functor which has a left adjoint which is also its right adjoint an equivalence ?

There are lots of examples. Here's what I think is in some sense the minimal one.

Let $C$ be the terminal category $mathbf{1}$ (one object, and only the identity arrow). Then for any category $D$, a left adjoint to the unique functor $G: D to mathbf{1}$ is an initial object of $D$, and a right adjoint is a terminal object. So, we're looking for a category $D$ that has a zero object (one that is both initial and terminal), but is not equivalent to the terminal category.

There are plenty of such categories $D$, e.g. $mathbf{Vect}$. But I guess the minimal one is the category $D$ generated by a split epimorphism. In other words, it consists of two objects, $0$ and $d$, and non-identity arrows

$$p: d to 0, i: 0 to d, ip: d to d,$$
satisfying $pi = 1_0$. Then $0$ is a zero object but $D$ is not equivalent to the terminal category.

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