I'm trying to learn some of the basics of category theory, and am stuck on this exercise (from Benjamin Pierce's text):
Show that in a poset considered as a category, the only equalizers are the identity arrows.
Here's an example of where I get confused:
Suppose we have a poset $P =(\{1,2\}, \leq)$ as a category, with arrows $1 \to 1$, $1 \to 2$, and $2 \to 2$. If this claim is true, then $e \colon 1 \to 2$ is not an equalizer of $f,g \colon 2 \to 2$. But how does it not fit the definition of an equalizer? We have $f \circ e = g \circ e$, and $e' \colon 1 \to 2$ is the only arrow such that both $f \circ e' = g \circ e'$ and there is a (unique) arrow $k \colon 1 \to 1$ with $e \circ k = e'$.
I figure I must be misunderstanding something simple, or have setup a bad example, and would really appreciate any help or pointers. Thanks!
(The text gives the following definition of an equalizer, which is where I got the names.)
An arrow $e \colon X \to A$ is an equalizer of a pair of arrows $f \colon A \to B$ and $g \colon A \to B$ if (1) $f \circ e = g \circ e$; and (2) whenever $e' \colon X' \to A$ satisfies $f \circ e' = g \circ e'$, there is a unique arrow $k \colon X' \to X$ such that $e \circ k = e'$.