I am self-studying Conceptual Mathematics by Lawvere and Schanuel. I tried Exercise 9 in Article II as follows, with my attempt included below. I just think it just does not seem to be what they are looking for.
Suppose $r$ is a retraction of $f$ (equivalently $f$ is a section of $r$) and let $e = f \circ r$. Show that $e$ is an idempotent. (As we’ll see later, in most categories it is true conversely that all idempotents can ‘split’ in this way.) Show that if $f$ is an isomorphism, then $e$ is the identity.
$$ let\ e=f\circ\ r\rightarrow \\ (f\circ\ r)\circ(f\circ\ r)=f\circ\ r\rightarrow\ \\ I_A\circ\ I_A=I_A\rightarrow\ \ \\ \ e\circ\ e=e $$
I tried checking Math StackExchange for this question but could not find it. Any help is much appreciated.
Because $r$ is a retraction of $f$, then $rf=\mathrm{id}$. It follows that $$ee = (fr)(fr) = f(rf)r = f(\mathrm{id})r = fr = e$$ so $e$ is idempotent.
If $f$ is an isomorphism, then $r = r(\mathrm{id}) = r(ff^{-1}) = (rf)f^{-1} = f^{-1}$, so $e=fr=ff^{-1} =\mathrm{id}$.
What you write seems like nonsense to me.