  The derived bracket of a Maurer-Cartan element in a differential graded Lie
algebra (DGLA) is well-known to define a differential graded Leibniz algebra.
It is also well-known that a Lie infinity morphism between DGLAs maps a
Maurer-Cartan element to a Maurer-Cartan element. Given a Lie-infinity
morphism, a Maurer-element and its image, we show that both derived
differential graded Leibniz algebras are related by a Leibniz-infinity
morphism, and we construct it explicitely. As an application, we recover a
well-known formula of Dominique Manchon about the commutator of the
star-product.
  Keywords: Leibniz algebras, Lie-infinity algebras, formality and
quantization.
