TITLE: How is a preorder a thin category?
QUESTION [7 upvotes]: I'm reading Category Theory for Programmers. In the chapter Categories Great and Small, it says:

A preorder is a category where there is at most one morphism going
  from any object a to any object b. Another name for such a category is
  “thin.” A preorder is a thin category.

My question is, if I have a <= b, a <= c, b <= d, c <= d like below:

doesn't that mean I have two morphisms going from object a to object d (i.e. a -> b -> d and a -> c -> d)?
What am I misunderstanding here?

REPLY [4 votes]: You did not mention how we turn a preorder (a set $S$ with a reflexive transitive relation ${\le}\subseteq S\times S$) into a category $\mathcal C$. Once you make this explicit, the thin-ness gets clear: 

The class of objects is the set $S$
The class of morphisms is the relation $\le$, where $a\le b$ (i.e., the pair $(a,b)\in S\times S$) is a morphism from $a$ to $b$.

Apparently, as we deal with a relation, there either is or is not a morphism (ond only one) from one object to another, i.e., $\mathcal C$ is certainly thin. (That it is a category to begin with follows from the preorder properties: reflexivity gives us identities, transitivity gives us composition)