Are not the real numbers a concrete category? I'm reading Awodey's Category Theory book. He says that the real numbers $\mathbb{R}$, regarded as a poset category, is small but not concrete; and that Pos is concrete but not small. However, every small category is concrete via its Cayley representation. In what way the author says that $\mathbb{R}$ is not concrete and Pos is?
 A: On page 7, Awodey explains the term "concrete category" as follows.

The categories that we have been considering so far are examples of what
are sometimes called concrete categories. Informally, these are categories in
which the objects are sets, possibly equipped with some structure, and the
arrows are certain, possibly structure-preserving, functions ...

This is the notion of concreteness that Awodey uses to in the paragraph you're asking about.
The category $\mathbb{Pos}$ is a concrete category because its objects are structured sets (posets), and its morphisms are structure preserving (in this case monotone) functions.
$\mathbb R$ is not a concrete category because its morphisms are not functions but abstractly derived from an order relation.
A: Keep in mind that a concrete category is a category $C$ together with a faithful functor $F : C \to Sets$.
The category of posets has an obvious forgetful functor that forgets the partial order and remembers only the set. This makes it into a concrete category.
$\mathbb{R}$ as a poset does not have an obvious forgetful functor.
However, it turns out that every poset can be made into a concrete category.
Let $(P, \leq)$ be a poset (or more generally, a preorder). Then consider the functor $F : P \to Sets$ sending all objects to $1$ and all arrows to $1_1$. This is a faithful functor.
So $\mathbb{R}$ can be made into a concrete category, but not in an especially natural or insightful way.
