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\$$ .