# Precise meaning of "diagram commutes" in a category theory?

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?

• Related question May 7 '18 at 8:49
• Composition (by definition) always gives a unique result. May 7 '18 at 8:50
• Also have a look at wikipedia. May 7 '18 at 8:57

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.
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.
• @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.) May 7 '18 at 9:58