So after working that out, I think the inverse functor is just the dual functor for a linear map.

If we have two different vector spaces \\(x \text{ and } y \text{ where } Ax =y\\) Since A is invertible, \\(A^{-1}A = I\\).

Then

\[x^\ast x = x^\ast I x = x^\ast (A^{-1}A) x = (x^\ast A^{-1})(Ax) = (Ax)^\ast(Ax) =y^\ast y\]

In terms of string diagrams maybe something like this?

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

Seems like something is missing though...

Edit: I have labeled the top circle \\(A^{-1}\\) but keeping with notation I should have labeled it just \\(A\\) .

If we have two different vector spaces \\(x \text{ and } y \text{ where } Ax =y\\) Since A is invertible, \\(A^{-1}A = I\\).

Then

\[x^\ast x = x^\ast I x = x^\ast (A^{-1}A) x = (x^\ast A^{-1})(Ax) = (Ax)^\ast(Ax) =y^\ast y\]

In terms of string diagrams maybe something like this?

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

Seems like something is missing though...

Edit: I have labeled the top circle \\(A^{-1}\\) but keeping with notation I should have labeled it just \\(A\\) .