TITLE: Equalisers in a poset
QUESTION [0 upvotes]: I am studying on Goldblatt's "Topoi". I have never taken a course on category theory before, so this is all quite new to me.
Exercise 3.10.2 asks to show that in a poset, the only equalisers are the identity arrows.
An arrow $i: e \rightarrow a$ is said to be an equaliser of arrows $f,g: a \rightarrow b$ if $f \circ i = g \circ i$ and for any arrow $h: c \rightarrow a$ for which $f \circ h = g \circ h$, one has that there exists a unique arrow $k: c \rightarrow e$ such that $i \circ k=h$.
A poset is a set $P$ with a partial ordering, that is, for any $p, q \in P$, one has $p \le p$ and $p \le q, q \le k \implies p \le k$. Posets are clearly categories, specifically preorders: given any two objects $p$ and $q$ in $P$ one has at most one arrow between those two objects, depending whether or not $p \le q$.
I think I am getting lost somewhere in the definitions. I tried to tackle this exercise by first showing that the identity is an equaliser, then that no other equaliser can exist in a poset.
So, let $f$ and $g$ be two arrows in the poset category. To show that the identity is an equaliser, we first need to show that $f \circ id = g \circ id$. This is obviously true if $f$ and $g$ are identical arrows. This is not necessarily the case though, right? $f$ and $g$ only need to have the same $\text{dom}$ and the same $\text{cod}$. In my understanding, $f$ and $g$ are arrows $f, g: P \rightarrow P$, that do not necessarily have to be identical. After all, it is entirely possible that $p \le q$ and $p \le k$ for $k \neq q$.
I feel like I am not grasping something very fundamental here. Any help in understanding the problem would be appreciated.
Please note that I have already read this question and the relative answer, but I don't feel like it solves my problem, as the answer assumes $f$ and $g$ are identical.

REPLY [1 votes]: General point to add to @user1073574's specific answer.
When starting out on any new area of maths it is always a good plan to have more than one textbook on the go (with a bit of luck, something that leaves you a bit puzzled in one book will be covered in a slightly different way and resolve the issue).
For an entry-level book on category theory as accessible as Topoi there is, for example, Harold Simmons’s book An Introduction to Category Theory. Or you can freely download my work-in-progress Beginning Category Theory from https://logicmatters.net/categories.
As it happens, your very question is answered in BCT at p. 101.
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