Commutative diagram

Remarks

Example: the square with vertices $A, B, C, D$ and edges $f \colon A \to B$, $g \colon B \to D$, $h \colon A \to C$, $k \colon C \to D$ commutes exactly when $g \circ f = k \circ h$. For instance, in the category of sets with $A = B = C = D = \R$, the maps $f = k = (x \mapsto 2x)$ and $g = h = (x \mapsto 3x)$ give a commuting square, since both composites are $x \mapsto 6x$. Non-example: with $f = k = (x \mapsto x + 1)$ and $g = h = (x \mapsto 2x)$, the square does not commute. Indeed $g(f(x)) = 2x + 2$ while $k(h(x)) = 2x + 1$. In practice one only asks for commutativity of the paths that the text names. For a diagram with a loop $f$, the convention above would force $f = f \circ f$. More formally, a diagram of shape $J$ is a functor from a category $J$ to $\mathcal{C}$.