  We consider an $A$-linear stable infinity-category $\mathcal{C}$ and the pair
$(\mathcal{HH}^\bullet(\mathcal{C}/A),\mathcal{HH}_\bullet(\mathcal{C}/A))$ of
the Hochschild cohomology spectrum (Hochschild cochain complex) and the
Hochschild homology spectrum (Hochschild chain complex). The purpose of this
paper is to provide a moduli-theoretic interpretation of the algebraic
structure on the Hochschild pair of $\mathcal{C}$. The algebraic structure on
the Hochschild pair is encoded by means of a two-colored topological operad
called Kontsevich-Soibelman operad. The notions of cyclic deformations and
equivariant deformations (of the Hochschild chain complex) associated to
deformations of $\mathcal{C}$ play a central role.
