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\\)?

> 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\\)?