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:
(s\circ r)\circ (s\circ r) &= s\circ (r\circ s) \circ r\\\\
&= s\circ 1\_x\circ r \\\\
&= s\circ r. \\\\
It's fun to think about what this means in various categories.