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\\).

**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\\).