  We show that the mapping cone of a morphism of differential graded Lie
algebras $\chi\colon L\to M$ can be canonically endowed with an
$L_\infty$-algebra structure which at the same time lifts the Lie algebra
structure on $L$ and the usual differential on the mapping cone. Moreover, this
structure is unique up to isomorphisms of $L_\infty$-algebras. The associated
deformation functor coincides with the one introduced by the second author in
arXiv:math.AG/0507287.
