There is an adjunction
This is (Quillen, prop. 4.4).
There is a standard structure of a category of weak equivalences? on both these categories, hence there are corresponding homotopy categories. (See also at model structure on simplicial Lie algebras and model structure on dg-Lie algebras.) The following asserts that the above adjunction is compatible with this structure.
of 1-connected objects on both sides.
This is in the proof of (Quillen, theorem. 4.4).
An early account is in part I, section 4 of
Graham Ellis, Homotopical aspects of Lie algebras Austral. Math. Soc. (Series A) 54 (1993), 393-419 (web)
İ. Akça and Z. Arvasi, Simplicial and crossed Lie algebras Homology Homotopy Appl. Volume 4, Number 1 (2002), 43-57.