TITLE: What is a Homotopy between $L_\infty$-algebra morphisms II
QUESTION [9 upvotes]: I would like to continue on question I asking, what is a homotopy between 
Lie infinity algebras, since I'm not satisfied in two directions:
1.) The naive approach to define a homotopy would be ('naive' in my opinion 
of course) the following:
Let $(L,(l)_{k\in\mathbb{N}})$ and $(M,(d)_{k\in\mathbb{N}})$ be two Lie infinity
algebras, let $f_\infty,g_\infty:L \to M$ be two morphism 
(in the most general sense) and let $(C(L),Q_L)$ and $(C(M),Q_M)$ be the 
appropriate differential graded coalgebras with induced morphism 
$F,G:C(L)\to C(M)$. 
Then a homotopy between $F$ and $G$ is a degree $+1$ map
$H:C(L)\to C(M)$ such that $F-G = HQ_L \pm Q_MH$
let signs and additional structure of $H$ (linear, coalgebra, ...) aside for
a moment.
It is strange, however, that I never saw this approach in the literature. Is this definition of homotopy equivalent to the previously mentioned approaches
in question I? 
Now for what is more important:
2.) The homotopy theory of Lie infinity algebras as given by Urs Schreiber in 
question I is obtained by 'transferring' the homotopy theory of differential graded Lie algebras 'along' an adjunction 
$F\dashv G$
($F:L_\infty Alg \to DGLA$ and $G:DGLA \to L_\infty Alg$)
Now the question that really irritates me for quite some time is: how can we be sure that we get the correct homotopy theory of Lie infinity algebras by transferring its 'shadow' in the category of DG Lie algebra back along the previous mentioned adjunction? To me it looks like we can not rule out that there is a more general definition of weak equivalences in the category of Lie infinity algebras, which just project under $F$ onto those we already know. 
Sorry if the second question is vague. 
Edit; I used the 'Lie algebra cohomology tag' since the homotopy theory of
Lie infinity algebras affects Lie algebra cohomology, too.

REPLY [2 votes]: First of all, you might want to look at the MO question How to define the equivalence of Maurer-Cartan elements in an $L_\infty$-algebra?, since of course this is effectively what you need (morphisms can be described as MC elements in a particular convolution $L_\infty$-algebra, I suspect you know that). 
Let me also say that some sort of approach to homotopy of the kind you are asking for is discussed by Martin Markl in this paper, and in this paper it is checked that the sort of definition proposed by Martin can be obtained from a Sullivan-type approach to homotopies (outlined in an answer to your first question) using appropriate homotopy transfer theorem for homotopy cooperads. 
It is worth remarking, in particular, that the definition of a "derivation homotopy" suggested in another answer to this question does not work: the crucial difference between $A_\infty$ and $L_\infty$ is that $A_\infty$ is a nonsymmetric operad, and therefore instead of the Sullivan's algebra of differential forms on the 1-simplex we can use the dg algebras of chains on the 1-simplex (which is noncommutative) - this gives the "derivation homotopy".