  In this paper we describe operads encoding two different kinds of
compatibility of algebraic structures. We show that there exist decompositions
of these in terms of black and white products and we prove that they are Koszul
for a large class of algebraic structures by using the poset method of B.
Vallette. In particular we show that this is true for the operads of compatible
Lie, associative and pre-Lie algebras.
