We show that morphisms from n homotopy unital $A_\infty$-algebras to a single one are maps over an operad module with n+1 commuting actions of the operad $A_\infty^\hu$, whose algebras are homotopy unital $A_\infty$-algebras. The operad $A_\infty$ and modules over it have two useful gradings related by isomorphisms which change the degree. The composition of $A_\infty^\hu$\n-morphisms with several entries is presented as a convolution of a coalgebra-like and an algebra-like structures.
Keywords: $A_\infty$-algebra, $A_\infty$-morphism, multicategory, multifunctor, operad, operad module, polymodule cooperad
2010 MSC: 18D50, 18D05, 18G35
Theory and Applications of Categories, Vol. 30, 2015, No. 46, pp 1552-1623.