Lax functoriality of Hochschild cochain complex

Unlike the Hochschild chain complex of an algebra, the Hochschild cochain complex of an algebra is not functorial. Nonetheless, we show that the Hochschild cochain complex of an algebra even a dg category is of lax functoriality, i.e., there exists a lax functor from bicategory of dg categories to bicategory of $B_\infty$-algebras which sends every dg category to its Hochschild cochain complex. This result is a homotopy version of the lax functoriality of center of an algebra obtained by Davydov, Kong, Runkel, Grady, Oren, et al, in the more general context of dg categories, and extends the restricted functoriality of Hochschild cochain complex of a dg category obtained by Keller to global lax functoriality.