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.

Is strange then 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 whats more important:

2.) The homotopy theory of Lie infinity algebras as given by Urs Schreiber in question I is obtained by 'transferring' the homotopy therory of differential graded Lie algebras 'along' the following adjunction:

We have the functor $R$ from Lie infinity algebras to DG Lie algebras, that projects the Lie infinity algebra onto the homology of its underlying chain complex and then forgets the higher brackets and on the other side we have the functor that includes a DG Lie algebra into Lie infinity algebras (because every DG Lie a is in particular a Lie infinity algebra). This gives the adjunction. Am I right here?

BUT, in general $R$ looses a lot of homotopy information! This was for example made clear in the work of Loday&Vallette on opards. In particular the complete homotopic information is transferred by the homotopy transfer theorem and $R$ is just a 'low degree shadow' of this, so to say.

Now the question that realy 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 $R$ onto those we already know.

Sorry if the second question is vague.