It is known that $f$, $g$, $h$ are isomorphisms.
It is known that $g\circ f = h^{-1}$.
I need to write down the proof of the following theorem. I am an amateur mathematician and am not an expert in category theory but write a book. The book shall contain this proof. Please help with wording the proof and base results and/or references to other books/articles I can use. (I am going to acknowledge you in my book.)
Theorem The diagram is commutative, every cycle in the diagram is an identity.
Related question: How do we prove commutativity of a diagram?