Associativity of composition

Statement

Let $f \colon A \to B$, $g \colon B \to C$ and $h \colon C \to D$ be functions. Then $$h \circ (g \circ f) = (h \circ g) \circ f.$$

Remarks

Both sides send each $a \in A$ to $h(g(f(a)))$, so they are equal. Because of this law, one can write $h \circ g \circ f$ without brackets. With the identity functions, the functions from a set to itself form a monoid under composition.

Explanations

L1 human

When three machines are chained, it does not matter which two you connect first. The input passes through $f$, then $g$, then $h$ in both cases.