Explicit Coequalizer in Set

Question

This is Problem 5.2.22 from Leinster's book.

Let $X$ be a set and $f: X \to X$ a map. Describe the coequalizer $\operatorname{coeq}(f,\operatorname{id}_X)$ of $f$ and $\operatorname{id}_X$ in $\mathbf{Set}$ as explicitly as possible.

Is it possible to describe it more explicitly than just $X/\sim$ where $\sim$ is generated by $x \sim f(x)$?

Answer 1

One way of describing the relation generated by $x ∼ f(x)$ is that $x∼y$ if and only if there is some $n,m ∈ ℕ$ such that $f^n(x) = f^m(y)$. You can see that by drawing $f$ as a quiver : here, each connected component looks like either an infinite tree or a loop with "arms" attached on it.

Answer 2

One interpretation would be that this coequaliser is the set of connected components of the graph whose vertex set is $X$ and whose edges are $\{\{x,f(x)\}:x\in X\}$. For fun you can try taking the coequaliser where $f\in S_n$, say $(1234)\in S_4$.