I've added some fun puzzles about things you can do in a compact closed category:

**Puzzle 284.** Using the cap and cup, any morphism \$$f \colon x \to y \$$ in a compact closed category gives rise to a morphism from \$$y^\ast\$$ to \$$x^\ast\$$. This amounts to 'turning \$$f\$$ around' in a certain sense, and we call this morphism \$$f^\ast \colon y^\ast \to x^\ast \$$. Write down a formula for \$$f^\ast\$$ and also draw it as a string diagram.

**Puzzle 285.** Show that \$$(fg)^\ast = g^\ast f^\ast \$$ for any composable morphisms \$$f\$$ and \$$g\$$, and show that \$$(1_x)^\ast = 1_x \$$ for any object \$$x\$$.

**Puzzle 286.** What is a slick way to state the result in Puzzle 285?

**Puzzle 287.** Show that if \$$x\$$ is an object in a compact closed category, \$$(x^\ast)^\ast\$$ is isomorphic to \$$x\$$.