  In this paper we prove, with details and in full generality, that the
isomorphism induced on tangent homology by the Shoikhet-Tsygan formality
$L_\infty$-quasi-isomorphism for Hochschild chains is compatible with
cap-products. This is a homological analog of the compatibility with
cup-products of the isomorphism induced on tangent cohomology by Kontsevich
formality $L_\infty$-quasi-isomorphism for Hochschild cochains.
  As in the cohomological situation our proof relies on a homotopy argument
involving a variant of {\bf Kontsevich eye}. In particular we clarify the
r\^ole played by the {\bf I-cube} introduced in \cite{CR1}.
  Since we treat here the case of a most possibly general Maurer-Cartan
element, not forced to be a bidifferential operator, then we take this
opportunity to recall the natural algebraic structures on the pair of
Hochschild cochain and chain complexes of an $A_\infty$-algebra. In particular
we prove that they naturally inherit the structure of an $A_\infty$-algebra
with an $A_\infty$-(bi)module.
