Given two homotopy algebras and an infinity-morphism between them, it is natural to ask that, if we can modify the two homotopy algebras in some structured way, can we modify the infinity-morphism in some similar way, so as to preserve the new structures? In this talk we describe a situation in which the answer is yes, and indicate how it is possible. We will also give an application of these results, to show that Tamarkin's construction of formality morphisms is equivariant with respect to the action of the Grothendieck-Teichmuller group.