  The amalgamated $T$-transform of a non-commutative distribution was
introduced by K.~Dykema. It provides a fundamental tool for computing
distributions of random variables in Voiculescu's free probability theory. The
$T$-transform factorizes in a rather non-trivial way over a product of free
random variables. In this article, we present a simple graphical proof of this
property, followed by a more conceptual one, using the abstract setting of an
operad with multiplication.
