Category theory: Enough that polygonal diagrams commute

I've read somewhere that for a categorical diagram to commute, it is enough that all its polygonal subdiagrams commute.

I want a reference and a detailed proof of this.

Please also give a formal definition of polygonal subdiagrams.

• Specifically pentagonal, or just polygonal? – Unit Jun 14 '15 at 14:39
• @Unit "pentagonal" was a typo. Corrected – porton Jun 14 '15 at 14:43
• Could you be more specific? It is true that in a monoidal category, if certain "basic" diagrams commute, then so do other more complicated diagrams constructed from them. For details see here – Matematleta Jun 14 '15 at 14:57
• @Chilango I don't believe this is the point at all. It's more like this: in a diagram like this, it's enough that each square commute for the whole diagram to commute. And for a square to commute it's enough that both triangles commute. And so on. – Najib Idrissi Jun 15 '15 at 8:05
• @porton You have read "somewhere"; where, exactly? And what is your precise definition, for a categorical diagram, of "to commute"? The answer will depend on the definition you have (because one could very well take "all polygonal subdiagrams commute" as a definition...). – Najib Idrissi Jun 15 '15 at 8:08