Why is the axiom $(h \circ g)\circ f=h\circ(g\circ f)$ used to define morphisms?

Why is the axiom $(h \circ g)\circ f=h\circ(g\circ f)$ used to define
morphisms?

We know that $hom(A,B)$ is a set of morphisms from $A$ to $B$. If $f\in
hom(A,B), g\in hom(B,C)$ and $h\in hom(C,D)$, then they have to satisfy
the following axiom:
$$(h \circ g)\circ f=h\circ(g\circ f)$$
Why is this axiom even mentioned? If $f,g,h$ are indeed mappings in the
traditional sense, then both are trivially equal to $h(g(f(a)))$, where
$a\in A$. Are there some morphisms which do not satisfy such a
relation/are not mappings in the usual sense?
Thanks in advance!