Are there any interesting examples of idempotent automorphisms $f:X\to X$ besides the identity, where $X$ is a structured space of some sort (or even just a class)?
My intuition says no since it seems that any function on a class $X$ which permutes a finite number of elements can't be idempotent unless it's the identity, but I'm not sure with infinite examples. To see this, observe that $$f(x)=y\ \text{and}\ f(y)=z\implies f^2(x)=z,$$ so if $y\neq z$ then $f$ can't be idempotent, so $f$ would have to be trivial on part of $X$. Let $${\sf fix}(f)=\{x\in X:f(x)=x\}$$ denote the fixed points of $f$. If $f$ is idempotent then $f(X)\subseteq{\sf fix}(f)$ by the above argument, so it seems like (for anything besides the identity) we would need $f$ not to be injective so we can map all elements of $X\setminus{\sf fix}(f)$ onto one element $y\in {\sf fix}(f)$ and be good to go, but endomorphic examples are trivial to come up with (e.g. any constant function on a point).