  Consider a monoidal category which is at the same time abelian with enough
projectives and such that projectives are flat on the right. We show that there
is a $B_{\infty}$-algebra which is $A_{\infty}$-quasi-isomorphic to the derived
endomorphism algebra of the tensor unit. This $B_{\infty}$-algebra is obtained
as the co-Hochschild complex of a projective resolution of the tensor unit,
endowed with a lifted $A_{\infty}$-coalgebra structure. We show that in the
classical situation of the category of bimodules over an algebra, this newly
defined $B_{\infty}$-algebra is isomorphic to the Hochschild complex of the
algebra in the homotopy category of $B_{\infty}$-algebras.
