>**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.

\\(y^\ast \stackrel{\sim}{\to} y^\ast \otimes 1 \stackrel{1_y \otimes \cap_x}{\to} y^\ast \otimes (x \otimes x^\ast) \stackrel{1_y \otimes f \otimes 1_x}{\to} y^\ast \otimes (y \otimes x^\ast) \stackrel{\sim}{\to} (y^\ast \otimes y) \otimes x^\ast \stackrel{\cup_y \otimes 1_x}{\to} 1 \otimes x^\ast \stackrel{\sim}{\to} x^\ast\\)

So the composite function would be \\( f^\ast = (1_y \otimes \cap_x)(1_y \otimes f \otimes 1_x)(\cup_y \otimes 1_x)\\).


Simply put if you pull the strings, the function gets flipped around.