Hey Anindya,

> I'm thinking we could perhaps use these tricks to "reshuffle" the inputs and outputs of any profunctor, ie any \$$\Phi : X \nrightarrow Y\$$ can be written as a profunctor \$$X \times Y^\text{op} \nrightarrow \textbf{1}\$$, or \$$\textbf{1} \nrightarrow X^\text{op} \times Y\$$, or \$$Y^\text{op} \nrightarrow X^\text{op}\$$, and we get these by judiciously composing with cup and cap...

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\$$"...