Function Composition has the property of distributivity:
$$(f\star g)\circ h = (f\circ h)\star(g\circ h)\;\forall f,g,\star \in\{+,-,\times,\div\}$$
I was wondering if these properties uniquely define composition.
Intuitively, this makes sense. For example:
$$(x\mapsto x^2)\circ f = (I\times I)\circ f = I\circ f \times I\circ f = f^2$$
and a similar process could be defined for any function.
But does this work when functions cannot be easily defined in terms of elementary operations?
Your conjecture is wrong. Let $q\colon A\to A$ be any map and define $f\odot g=f\circ q\circ g$. Then $$ (f\star g)\odot h=(f\star g)\circ (q\circ h)=f\circ(q\circ h)\star g\circ(q\circ h)=f\odot h\star g\odot h$$