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In Hatcher's Algebraic Topology, on page 246 (here, in the book), about two-thirds down the page he states that showing the commutativity of the two squares shown in the diagram, not involving the boundary or coboundary maps, is a 'triviality'.

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I can sort of see how it works, but I simply can't work out the details satisfactorily, and all the while the word 'triviality' has been corroding my peace-of-mind like acid rain on a gargoyle. I don't suppose somebody could help me out?

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