  Various compatibility conditions among replicated copies of operations in a
given algebraic structure have appeared in broad contexts in recent years.
Taking an uniform approach, this paper gives an operadic study of compatibility
conditions for nonsymmetric operads with unary and binary operations, and
homogeneous quadratic and cubic relations. This generalizes the previous
studies for binary quadratic operads. We consider three compatibility
conditions, namely the linear compatibility, matching compatibility and total
compatibility, with increasingly strict restraints among the replicated copies.
The linear compatibility is in Koszul dual to the total compatibility, while
the matching compatibility is self dual. Further, each compatibility can be
expressed in terms of either one or both of the two Manin square products.
