Matthew wrote:

> Do you think we can recover all of the "yanking conditions" of a [symmetric compact closed monoidal category](https://en.wikipedia.org/wiki/Compact_closed_category#Symmetric_compact_closed_category)?

> $A\xrightarrow{\cong} A\otimes I\xrightarrow{A\otimes\eta} A\otimes (A^\ast\otimes A) \xrightarrow{\cong} (A\otimes A^\ast) \otimes A \xrightarrow{\varepsilon\otimes A} I\otimes A\xrightarrow{\cong} A \\\\ A^\ast\xrightarrow{\cong} I\otimes A^\ast\xrightarrow{\eta\otimes A^\ast}(A^\ast\otimes A)\otimes A^\ast\xrightarrow{\cong} A^\ast\otimes (A\otimes A^\ast)\xrightarrow{A^\ast \otimes\varepsilon} A^\ast\otimes I\xrightarrow{\cong} A^\ast$

> Here "\$$\ast\$$" would be "\$$\mathrm{op}\$$", "\$$\otimes\$$" would be "\$$\times\$$", and "\$$\to\$$" would be "\$$\nrightarrow\$$

Yes! And \$$\eta\$$ is the cap, and \$$\varepsilon\$$ is the cup.

This will be the subject of my next lecture, and you folks have done most of the work for me!

I call the yanking identities the **zig-zag equations.** If we draw a cap like a cap, a cup like a cup, and an identity like a pipe, they say this: