  The formality morphism $\boldsymbol{\mathcal{F}}=\{\mathcal{F}_n$,
$n\geqslant1\}$ in Kontsevich's deformation quantization is a collection of
maps from tensor powers of the differential graded Lie algebra (dgLa) of
multivector fields to the dgLa of polydifferential operators on
finite-dimensional affine manifolds. Not a Lie algebra morphism by its term
$\mathcal{F}_1$ alone, the entire set $\boldsymbol{\mathcal{F}}$ is an
$L_\infty$-morphism instead. It induces a map of the Maurer-Cartan elements,
taking Poisson bi-vectors to deformations $\mu_A\mapsto\star_{A[[\hbar]]}$ of
the usual multiplication of functions into associative noncommutative
$\star$-products of power series in $\hbar$. The associativity of
$\star$-products is then realized, in terms of the Kontsevich graphs which
encode polydifferential operators, by differential consequences of the Jacobi
identity. The aim of this paper is to illustrate the work of this algebraic
mechanism for the Kontsevich $\star$-products (in particular, with harmonic
propagators). We inspect how the Kontsevich weights are correlated for the
orgraphs which occur in the associator for $\star$ and in its expansion using
Leibniz graphs with the Jacobi identity at a vertex.
