I'm trying to understand a part of the PhD thesis of Kenji Lefèvre-Hasegawa (e.g. available here). My question is about the proof of Lemma 1.3.2.3b stating:

Remarquons que nous avons un isomorphisme de complexes $\bigoplus_{i\geq 1}Gr_i(\Omega BA)\to \Omega \overline{T^c}V$ qui identifie à la composante $Gr_i(\Omega BA), i\geq 1$, à la somme des $S^{-1} V^{\otimes i_1}\otimes\dots\otimes S^{-1} V^{\otimes i_k}\subset (S^{-1} \overline{T^c}V)^{\otimes k}$, où $k\geq 1$ et où $i_1+\dots+i_k=i$.

Here $V=SA$, where $S$ is the shift functor. $A$ is a dg algebra and $B$ and $\Omega$ are the bar and cobar resolution, respectively, i.e. $BA=\bigoplus_{i\geq 1} V^{\otimes i}$ (with the coproduct $\Delta: V^{\otimes i}\to \bigoplus_{p+q=i} V^{\otimes p}\otimes V^{\otimes q}$ and the filtration $BA_{[i]}=ker\Delta^{(i)}$) and $\Omega BA=\bigoplus_{i\geq 1}S^{-1}BA^{\otimes i}$ with the filtration induced by that of $BA$.

Can someone give more details?