How do we describe the 'commutativity' of two different operators/operations?

Question

For example the defintion of a conjugate transpose/Hermitian transpose of a matrix. I can either take it's conjugate and then transpose it or the other way.

${\displaystyle {\boldsymbol {A}}^{\dagger }}={\displaystyle {\boldsymbol {A}}^{\mathrm {H} }=\left({\overline {\boldsymbol {A}}}\right)^{\mathsf {T}}={\overline {{\boldsymbol {A}}^{\mathsf {T}}}}}$

Wikipedia: Conjugate_transpose

Answer 1

Let $f(B):=B^H, g(B):=\overline{B}$ and $h(B):= B^T.$

Then

$$f(A)=h(g(A))=g(h(A)).$$

Hence

$$f=h \circ g = g \circ h.$$