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?
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?
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\\) .
Version 2.1.8p2
