Let $f : X \to Y$ and $g : Y \to Z$ and $h : Z \to W$ each be a D18: Map.

By definition, both compositions are maps $X \to Z$, so it remains to show that the values of both compositions coincide on each element of $X$. If $x \in X$, then \begin{equation} \begin{split} (h \circ (g \circ f))(x) = h((g \circ f)(x)) & = h(g(f(x))) \\ & = (h \circ g)(f(x)) = ((h \circ g) \circ f)(x) \end{split} \end{equation} Hence, the claim is now a consequence of R1869: Characterisation of equality of maps. $\square$