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On page 173 of Paolo Aluffi's Algebra: Chapter 0 he says:

Fiber diagrams are commutative, but "even better" than commutative.

What is he referring to when he makes the statement that they are even better than commutative? The only thing I can think of is that these types of diagrams have a universal property, though I don't see why this property would be emphasized with a square in the center of the diagram to denote it. Is this what he means, or is there something else special about diagrams?

For Context:

enter image description here

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I denote pullback with a different symbol than a square in it but yes that's just what it means. –  user51427 Dec 10 '12 at 19:07
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up vote 4 down vote accepted

You are right: he means that it has the universal property indicated in the first diagram, and the square in the middle is just a shorthand to denote this fact.

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