Describe the equalizer of $f,1:X\to X$ as explicitly as possible Exercise 5.2.22 from Leinster asks to describe the equalizer of $f,1:X\to X$ in $\mathbf{Set}$, where $f:X\to X$ is a map, as explicitly as possible.
What level of explicitness is expected? By definition, it is a pair $(S,h)$ where $S$ is a set and $S\to X$ is a map such that $fh=h$, and if $(S'g)$ is another such pair, then there is a unique $\bar g:S'\to S$ such that $h\bar g=g$. How can one make this more explicit?

The equalizer of $s,t:X\to Y$ in $\mathbf{Set}$ is $(E,i)$ where $E=\{x\in X:s(x)=t(x)\}$ and $i:E\to X$ is the inclusion map. This is clearly a fork, and if $(A,g)$ is another fork, then define $\bar g:A\to E, a\mapsto g(a)$. Note that $g(a)$ indeed lives in $E$ because $sg=tg$, and $\bar g$ is the unique map $A\to E$ such that $i\bar g=g$.
Apply this to $Y=X,\ s=1,\ t=f$.
