Fredrick wrote in comment #19:

> I presume that the unique category with a single object, \$$\star\$$, and one morphism, \$$1_{\star}\$$ is a preorder.

Yes.

A preorder is a category where there's at most one morphism from any object to any object. So, the smallest category that's _not_ a preorder will have one object, say \$$\star\$$, and _two_ morphisms from that object to itself.

What are these morphisms like? One of them must be the identity \$$1\_\star : \star \to \star\$$. The other is something else, call it \$$f : \star \to \star.\$$

We must have

$1\_\star \circ 1_\star = 1\_\star$

and we must have

$f \circ 1\_\star = f= 1\_\star \circ f .$

So, the only question is: what's \$$f \circ f \$$? Apparently there are two possibilities: \$$1\_\star\$$ and \$$f\$$.

**Puzzle.** Why did Dan Oneata choose \$$f \circ f = 1\_\star\$$?