**Puzzle 102.** What is the category with the fewest morphisms that is not a preorder?

Dan Oneata gave a nice answer in comment #3: he chose the unique category with a single object, \$$\star\$$, and two morphisms, \$$1_{\star}\$$ and \$$f\$$, such that \$$f \circ f = 1\_\star\$$.

**Puzzle 102\$${}^\prime\$$.** What's the usual name for this category?

Hint: a category with one object is the same as a [monoid](https://en.wikipedia.org/wiki/Monoid), and a monoid where every element has an inverse is called a [group](https://en.wikipedia.org/wiki/Group_(mathematics)).