>**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)\$$.

![function_cap_cup](http://aether.co.kr/images/function_cap_cup.svg)

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