How is "at most one arrow" used to proof a special category gives rise to a preorder set?

Question

I'm working on exercise 2 on page 1 of Basic Category Theory By Jaap van Oosten.

The problem is:

If $\mathcal{C}$ is a category such that $\mathcal{C}_0$ is a set, and such that for any two objects $X, Y$ of $\mathcal{C}$ there is at most one arrow: $x \rightarrow y$, then $\mathcal{C}_0$ is a preorder set.

I try to prove this by using the definition of preorder set:

• element: all objects in $\mathcal{C}$
• binary relation: $f: a \rightarrow b$ is corresponding to $a \le b$.
• reflexivity: implied by identity morphisms of $\mathcal{C}$.
• transitivity: implied by composition of $\mathcal{C}$.

It seems the condition at most one arrow is not used anywhere. Does my proof work? Am I missing something?

Answer 1

The problem is in the statement : any set is a preorder (for example declare all the elements to be minimal).

The correct statement of the exercise should be as follow (at least I guess this is what the author means) :

Let $\mathsf{Pos}$ be the category whose objects are posets and whose morphisms are non decreasing maps. Let $\mathsf{Slim}$ be the category whose objects are the categories $\mathcal C$ such that $\mathcal C_0$ is a set and that for every $x,y \in \mathcal C_0$ there is at most one arrow $x \to y$ and whose morphisms are the functors between such categories.

Then show that $\mathsf{Pos}$ is equivalent to $\mathsf{Slim}$.