I would like to continue on [question I][1] 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][2]? 

Now whats more important:

2.) The homotopy theory of Lie infinity algebras as given by Urs Schreiber in 
[question I][3] 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][4] 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. 


  [1]: http://mathoverflow.net/questions/139175/what-is-a-homotopy-between-l-infty-algebra-morphisms "Question I"
  [2]: http://mathoverflow.net/questions/139175/what-is-a-homotopy-between-l-infty-algebra-morphisms "Question I"
  [3]: http://mathoverflow.net/questions/139175/what-is-a-homotopy-between-l-infty-algebra-morphisms "Question I"
  [4]: http://math.unice.fr/~brunov/Operads.pdf