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

Compatibility with cap-products in Tsygan's formality and homological Duflo isomorphism