In Steve Awodey's Category Theory, 2nd ed. (2010), in the section on generalized elements when he's discussing global elements he says that, with posets, arrows $1 \rightarrow P$ correspond to the elements of the underlying set of the poset $P$ (p. 36). He then goes on to say that in any category with a terminal object $1$, arrows $1 \rightarrow A$ are the points or global elements of $A$.
On the next page (36), he compares two posets $X = \{x \leq y, x \leq z\}$ and $A = \{a \leq b \leq c\}$ and says that they have the same number of global elements, the three elements of the sets (presumably he means $\{x, y, z\}$ and $\{a, b, c\}$?).
Now, my question concerns a poset like $A$ once "categorified". We then have the category $\mathbf{A}$ whose objects, I believe, are $a, b,$ and $c$. The morphisms of $\mathbf{A}$ would be $a \rightarrow a, a \rightarrow b, a \rightarrow c, b \rightarrow b, b \rightarrow c,$ and $c \rightarrow c$. From what I understand $c$ is the terminal object of $\mathbf{A}$ and so the global elements would be morphisms of the form $c \rightarrow x$ for $x\in A$. Clearly, there is only one such arrow in $\mathbf{A}$ (the identity morphism $c \rightarrow c$) and so it seems there is only one global element, not 3 corresponding to the elements of the underlying set of the poset. Or is it that global elements don't have to be contained in the relevant category and so there are three morphisms $c \rightarrow c, c \rightarrow b, c \rightarrow a$ which are the global elements, even though only one of them is actually in $\mathbf{A}$?
I'm clearly missing something fairly simple, presumably misunderstanding a definition or missing some distinction. Where am I going wrong?
You're confusing two very different situations: the category of posets, and a single specific poset considered as a category.
Elements of a poset $P$ correspond to morphisms $1\to P$ in the category of posets, where $1$ is the terminal object in the category of posets. That is, elements of $P$ (in the usual sense) are global elements of $P$, as an object in the category of posets.
On the other hand, in your example poset $A$, morphisms $c\to x$ are global elements of $x$ as an object of the category $A$. You're correct that such a global element exists only for $x=c$ (and there is only one such global element in that case). But this is totally different from talking about global elements of the object $A$ in the category of posets. In the case of morphisms $c\to x$, $A$ is the ambient category, and in the other case $A$ is the object (in the category of posets) which is the codomain of the global elements you are talking about (that is, it is playing the role of $x$).