  We study the deformation complex of a canonical morphism $i$ from the
properad of (degree shifted) Lie bialgebras $\mathbf{Lieb}_{c,d}$ to its
polydifferential version $\mathcal{D}(\mathbf{Lieb}_{c,d})$ and show that it is
quasi-isomorphic to the oriented graph complex
$\mathbf{GC}^{\text{or}}_{c+d+1}$, up to one rescaling class. As the latter
complex is quasi-isomorphic to the original graph complex $\mathbf{GC}_{c+d}$,
we conclude that the space of homotopy non-trivial infinitesimal deformations
of the canonical map $i$ can be identified with the Grothendieck-Teichm\"uller
Lie algebra $\mathfrak{grt}$; moreover, every such an infinitesimal deformation
extends to a genuine deformation of the canonical morphism $i$ from
$\mathbf{Lieb}_{c,d}$ to $\mathcal{D}(\mathbf{Lieb}_{c,d})$. The full
deformation complex is described with the help of a new graph complex of so
called entangled graphs, whose suitable quotient complex is shown to contain
the tensor product $H(\mathbf{GC}_c) \otimes H(\mathbf{GC}_d)$ of cohomologies
of Kontsevich graph complexes $\mathbf{GC}_c \otimes \mathbf{GC}_d$.
