  Let $\g_1$ and $\g_2$ be two dg Lie algebras, then it is well-known that the
$L_\infty$ morphisms from $\g_1$ to $\g_2$ are in 1-1 correspondence to the
solutions of the Maurer-Cartan equation in some dg Lie algebra
$\Bbbk(\g_1,\g_2)$. Then the gauge action by exponents of the zero degree
component $\Bbbk(\g_1,\g_2)^0$ on $MC\subset\Bbbk(\g_1,\g_2)^1$ gives an
explicit "homotopy relation" between two $L_\infty$ morphisms. We prove that
the quotient category by this relation (that is, the category whose objects are
$L_\infty$ algebras and morphisms are $L_\infty$ morphisms modulo the gauge
relation) is well-defined, and is a localization of the category of dg Lie
algebras and dg Lie maps by quasi-isomorphisms. As localization is unique up to
an equivalence, it is equivalent to the Quillen-Hinich homotopical category of
dg Lie algebras [Q1,2], [H1,2]. Moreover, we prove that the Quillen's concept
of a homotopy coincides with ours. The last result was conjectured by
V.Dolgushev [D].
