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It's weird, but none of introductory category theory books I've seen so far provide rigor definition of "diagram commutes" saying, however they do use such a term extensively.

As far as I was able to deduce from a context, one says that diagram commutes iff there always exists exactly one arrow satifsying somewhat composition. Let's take natural transformation as a simple example: $$G(f) \circ \alpha_A = m = \alpha_{A'} \circ F(f)$$ By saying that such a "square commutes" it is meant that there is unique morphism $m$ equal to the specified compositions.

Is it correct?

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  • $\begingroup$ Related question $\endgroup$
    – drhab
    May 7 '18 at 8:49
  • $\begingroup$ Composition (by definition) always gives a unique result. $\endgroup$ May 7 '18 at 8:50
  • $\begingroup$ Also have a look at wikipedia. $\endgroup$
    – drhab
    May 7 '18 at 8:57
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You're basically right. A diagram commutes iff every composition of arrows from a given object $X$ to a given object $Y$, via any number of intermediate steps, is the same morphism.

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    $\begingroup$ With the caveat that this usually only requires paths of length at least two to satisfy this (so it does not apply to parallel arrows). And in some more complicated diagrams, if one includes an arrow from a set of parallel ones, no other paths can include other of those arrows (so essentially one has written several different diagrams in one). $\endgroup$ May 7 '18 at 8:49
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The archetypal commutative diagram is the following: Take four objects $X_1, X_2, Y_1, Y_2$ and morphisms $f_i:X_i\to Y_i$ as well as $g_X:X_1\to X_2, g_Y:Y_1\to Y_2$, i.e. a diagram like this: $$ \require{AMScd} \begin{CD} X_1 @>{g_X}>> X_2 \\ @Vf_1VV @Vf_2VV \\ Y_1 @>>g_Y> Y_2 \end{CD} $$ Then this diagram commutes iff $f_2\circ g_X = g_Y\circ f_1$. In other words, starting at $X_1$, "go right" and "go down" commutes.

The general idea is the same: A diagram commutes iff all possible paths (using the arrows explicitly drawn in the diagram) from one object to another object result in the same morphism. There may be caveats, as detailed by @Tobias in a comment to the other answer.

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    $\begingroup$ I would say the archetypal commuting diagram is a triangle. You can always cut up a big diagram in triangles, not always in squares. $\endgroup$ May 7 '18 at 9:03
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    $\begingroup$ @NajibIdrissi I agree that the triangle is a more fundamental diagram. However, as the (in my opinion) likely origin of the phrase "commutative", and as explanation of why we use that phrase, I think the square is the archetypal example. (Archetypal $\neq$ fundamental.) $\endgroup$
    – Arthur
    May 7 '18 at 9:58

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