We can prove by definition of transposition and inversion that for an inversible matrix A, $(A^T)^-1 = (A^{-1})^T$
I would like to see through other means when this can happen (for other matrix functions)
In this case, for example, let's define a Subspace of $R^{n \times n}$ and two functions:
$ S = {A \in R^{n \times n} / \exists A^{-1}} $
$f:S \rightarrow S$, $f(A) = A^T$
$g:S \rightarrow S$, $g(A) = A^{-1}$
These functions are not just isomorphisms but are also their own inverse.
Is this enough to state (fog)(A) = (gof)(A)?
If this is the case, how do we call an isomorphism that is its own inverse?
Being an isomorphism that is its own inverse is not sufficient. Let me define on $2\times2$ matrices such an isomorphism $\phi$ which simply changes the sign of the $(1,2)$ entry.
Now $\phi(A)^{-1}$ and $\phi(A^{-1})$ are different