Hey Anindya,

> I'm afraid I've only got the vaguest understanding of how coherence conditions work in monoidal categories, Matthew.

Hmm... I think I am in the same boat.

(EDIT: n/m, you've got more muscle than me...)

Maybe we can start with something easier. I bet we could prove a consequence of the category being symmetric compact closed.

On page 42 of John Baez and Mike Stay's [Rosetta Stone (2009)](http://math.ucr.edu/home/baez/rosetta.pdf) they define:

\[ Y \multimap Z := Y^\ast ⊗ Z \]

They then write on the next page, approximately:

\[ \mathrm{Hom}(Y \otimes X, Z) \cong \mathrm{Hom}(X, Y^\ast \otimes Z) \]

(They're using *Linear Logic* rather than conventional category theory, but hopefully John can come by and correct me if I am making a grave mistake here.)

Based on this, perhaps we can recover \\((Y \times -) \dashv (Y^{\mathrm{op}} \times -)\\).

I just tried to show a relationship like this over in Lecture 66 for posets and preorders. I don't think it should be too hard to show this for profunctors (famous last words)...

> I'm afraid I've only got the vaguest understanding of how coherence conditions work in monoidal categories, Matthew.

Hmm... I think I am in the same boat.

(EDIT: n/m, you've got more muscle than me...)

Maybe we can start with something easier. I bet we could prove a consequence of the category being symmetric compact closed.

On page 42 of John Baez and Mike Stay's [Rosetta Stone (2009)](http://math.ucr.edu/home/baez/rosetta.pdf) they define:

\[ Y \multimap Z := Y^\ast ⊗ Z \]

They then write on the next page, approximately:

\[ \mathrm{Hom}(Y \otimes X, Z) \cong \mathrm{Hom}(X, Y^\ast \otimes Z) \]

(They're using *Linear Logic* rather than conventional category theory, but hopefully John can come by and correct me if I am making a grave mistake here.)

Based on this, perhaps we can recover \\((Y \times -) \dashv (Y^{\mathrm{op}} \times -)\\).

I just tried to show a relationship like this over in Lecture 66 for posets and preorders. I don't think it should be too hard to show this for profunctors (famous last words)...