 $i=2,\ldots, n-1$
\[
\xymatrix{
&&&\\
\Sigma Y'_{i-1} \ar[u]\ar[r] & \Sigma Y_{i-1}\ar[r]\ar[u]&\Sigma^{i-1} X_1\ar@{=}[d]\ar[r] &
\\
Y'_{i}\ar[u] \ar[r] & Y_{i}\ar[r]\ar[u]&\Sigma^{i-1} X_1\ar[r] &
\\
X_i\ar[u]\ar@{=}[r] &X_i\ar[u]\\
}
\]
with rows and columns that are distinguished triangles, where the maps not involving $Y'$'s are taken from the Postnikov system.  Hence similar to Remark \ref{lem:intuition}, $Y'_{i}$ has subquotients $X_{i},\Sigma X_{i-1},\ldots,\allowbreak\Sigma^{i-2}X_2$.

We have a distinguished triangle
\[
Y'_{n-1}\rightarrow Y_{n-1}\rightarrow \Sigma^{n-2} X_1\rightarrow 
\]
and hence  $\alpha$ is determined up to a composition $X_0\xrightarrow{\delta} \Sigma^{-n+2}
Y'_{n-1}\rightarrow \Sigma^{-n+2}Y_{n-1}$. Now  $ \Sigma^{-n+2}
Y'_{n-1}$ has subquotients $\Sigma^{-n+2}X_{n-1},\ldots,\Sigma^{-1}X_2$ and hence 
by \eqref{toda:coset} we obtain that any map $X_0\xrightarrow{\delta} \Sigma^{-n+2}Y'_{n-1}$  factors through $\Sigma^{-n+2}X_{n-1}$. Hence we obtain that $\alpha$ is determined up
to a composition $X_0\xrightarrow{\delta'} \Sigma^{-n+2}X_{n-1}\rightarrow \Sigma^{-n+2} Y'_{n-1}\rightarrow \Sigma^{-n+2} Y_{n-1}$ which by construction is the same as a composition
$X_0\xrightarrow{\delta'} \Sigma^{-n+2}X_{n-1}\xrightarrow{\Sigma^{-n+2}i} Y_{n-1}$. We now finish as for $\beta$.\qedhere
\end{enumerate}
\end{proof}
\subsection{Postnikov systems associated to twisted complexes}
In this section $\mathfrak{a}$ is an $A_\infty$-category.
\subsubsection{More on the category $\operatorname{Free}(\mathfrak{a})$}
Recall that in \S\ref{sec:free} we introduced the strict endo-functor  $\Sigma$ of $\operatorname{Free}(\mathfrak{a})$. 
Below we introduce some more notation concerning the category $\operatorname{Free}(\mathfrak{a})$. If $X=\Sigma^x Y$ for $Y\in \operatorname{Ob}(\mathfrak{a})$ then we let $\eta_{X,a,b}:
\Sigma^aX\rightarrow \Sigma^b X$ be given by $\sigma^{b-a}\operatorname{id}_Y$. Note that $m_2(\eta_{X,b,c},\eta_{X,a,b})=\eta_{X,a,c}$. All operations
on $\mathfrak{a}$, except $m_2$, vanish when one of its arguments is of the form $\eta_{X,b,c}$. 
Moreover we have formulas
\begin{align*}
m_n(\dots,f,m_2(\eta_{X,a,b},g),\ldots)&=m_n(\dots,m_2(f,\eta_{X,a,b}),g,\ldots)\\
m_n(m_2(\eta_{X,a,b},f),\dots)&=(-1)^{(-2+n)(b-a)}m_2(\eta_{X,a,b},m_n(f,\ldots))\\
m_n(\dots,m_2(g,\eta_{X,a,b})))&=m_2(m_n(\ldots,g),\eta_{X,a,b}).
\end{align*}
and their $b$-versions which are useful for computations
\begin{equation}
\label{eq:useful}
\begin{aligned}
b_n(\dots,sf,sm_2(\eta_{X,a,b},g),\ldots)&=(-1)^{b-a}b_n(\dots,sm_2(f,\eta_{X,a,b}),sg,\ldots)\\
b_n(sm_2(\eta_{X,a,b},f),\dots)&=m_2(\eta_{X,a,b},b_n(sf,\ldots))\\
b_n(\dots,sm_2(g,\eta_{X,a,b})))&=m_2(b_n(\ldots,sg),\eta_{X,a,b}).
\end{aligned}
\end{equation}
Below we usually write $\eta_{X,a,b}g$ for $m_2(\eta_{X,a,b},g)$ and similarly $m_2(g,\eta_{X,a,b})$. By the vanishing of $m_3$ on arguments involving $\eta_{X,a,b}$
this will not lead to confusion. Sometimes
we also write $\eta^{-1}_{X,a,b}$ for $\eta_{X,b,a}$. One verifies using
the definition of the functor $\Sigma$ (see \S\ref{sec:free})
that for $f:\Sigma^a X\rightarrow \Sigma^b Z$ one has
\begin{equation}
\label{eq:sigmadef}
\Sigma^nf=(-1)^{n|f|}\eta_{Z,b,b+n}f\eta_{X,a+n,a}.
\end{equation}
Finally we put $\eta_{X}=\eta_{X,0,1}$.
\subsubsection{More on the triangulated structure of $\operatorname{Tw} \mathfrak{a}$}
Let $f:(A,\delta_A)\rightarrow (B,\delta_B)$ be a closed morphism of degree $0$ in $\operatorname{Tw} \mathfrak{a}$. To $f$ we associate a triangle in $H^\ast(\operatorname{Tw} \mathfrak{a})$.
\begin{equation}
\label{eq:standard1}
(A,\delta_A) \xrightarrow{f} (B,\delta_B)\xrightarrow{i} (C(f),\delta_{C(f)}) \xrightarrow[(1)]{p} (A,\delta_A)
\end{equation}
where $C(f)=\Sigma A\oplus B$ and 
\[
\delta_{C(f)}=\begin{pmatrix}
\Sigma \delta_A &0\\
f\eta_A^{-1}&\delta_B
\end{pmatrix}
\]
and furthermore
\begin{equation}
\label{eq:iptwisted}
i=\begin{pmatrix}0\\{\operatorname {id}}_B
\end{pmatrix},\qquad 
p=(\eta_A^{-1}\ 0).
\end{equation}
The following lemma is an easy verification:
\begin{lemma} The triangles \eqref{eq:standard1} are distinguished according to Definition \ref{triangulatedstructure}.
\end{lemma}
\subsubsection{Postnikov systems from objects in $\operatorname{Tw} \mathfrak{a}$}
\begin{proposition}
\label{prop:postnikovtwisted}
A twisted complex in $\operatorname{Tw} \mathfrak{a}$
\[
Y_n=(\Sigma^nX_0\oplus \Sigma^{n-1}X_1\oplus\cdots\oplus X_n,\delta)
\]
with $X_i\in \operatorname{Free}(\mathfrak{a})$ gives rise to a Postnikov system in $H^\ast(\operatorname{Tw}\mathfrak{a})$ built on the complex
\[
X_0\xrightarrow{d_0}X_1\xrightarrow{d_1}\cdots\xrightarrow{d_{n-1}}X_n
\]
with 
\[
d_{j-1}=(-1)^{n-j} \eta_{X_j,n-j,0}\cdot\delta_{j,j-1} \cdot\eta_{X_{j-1},0,n-j+1},
\]
where $\delta_{j,j-1}:\Sigma^{n-j+1} X_{j-1}\rightarrow \Sigma^{n-j}X_j$ is the $(j,j-1)$ entry of the matrix $\delta$ (the $\cdot$'s are for easier reading).

In the Postnikov system we also have
\[
Y_j=(\Sigma^{j}X_0\oplus \Sigma^{j-1}X_1\oplus\cdots\oplus X_j,\delta_{Y_j})
\]