Keith - yes, that's a nice observation! You can prove this in any category, not just the category of sets. If \$$r: y \to x\$$ and \$$s: x \to y\$$ are morphisms in any category, and \$$r \circ s = 1_x\$$, then \$$s \circ r\$$ is idempotent:

\$\begin{array}{cl} (s\circ r)\circ (s\circ r) &= s\circ (r\circ s) \circ r\\\\ &= s\circ 1\_x\circ r \\\\ &= s\circ r. \\\\ \end{array} \$

It's fun to think about what this means in various categories.