We show that morphisms from n $A_\infty$-algebras to a single one are maps over an operad module with n+1 commuting actions of the operad $A_\infty$, whose algebras are conventional $A_\infty$-algebras. The composition of $A_\infty$-morphisms with several entries is presented as a convolution of a coalgebra-like and an algebra-like structures. Under these notions lie two examples of Cat-operads: that of graded modules and of complexes.
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. 45, pp 1501-1551.