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.

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Related question: How do we prove commutativity of a diagram?

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    $\begingroup$ Use induction. For big enough cycles a smaller cycle must appear as part of your composition (by the pigeonhole principle) . Contract by that cycle, and then use the induction hypothesis. $\endgroup$ – PVAL-inactive Aug 5 '15 at 19:44
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    $\begingroup$ In this case you would just need to explicitly check every cycle of length at most 3 is the identity. $\endgroup$ – PVAL-inactive Aug 5 '15 at 19:45
  • $\begingroup$ @PVAL But how to show that every cycle is composed from cycles of the length no more than 3? $\endgroup$ – porton Aug 5 '15 at 19:50
  • $\begingroup$ @PVAL Also this proves only that cycles are identities, but we need to prove also that the diagram is commutative $\endgroup$ – porton Aug 5 '15 at 19:52
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    $\begingroup$ Any composition of 3 length, hits one point twice (by the pigeonhole principle). So either this path is a cycle or it is a cycle composed with another composition. $\endgroup$ – PVAL-inactive Aug 5 '15 at 20:01

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