Repetitive Algebra. I am studying the category of finitely generated left modules over the repetitive algebra and I'm using the book of Happel: Triangulated categories in the representation theory of finite dimensional algebras. I have several questions about this category.
First, I will write the definition of this algebra.
Definition: Let $A$ be a $K$-algebra of finite dimension, where $K$ is an algebraically closed. Denote by $D=\operatorname{Hom}_K (-,K)$ the standard duality on $\mod A$. 
Thus, the underlying vectorspace of $\hat{A}$ is given by
$$
\hat{A} = \left(\bigoplus_{i \in \mathbb{Z}} A\right) \bigoplus \left(\bigoplus_{i \in \mathbb{Z}} DA\right).
$$
We denote the elements of $\hat{A}$ by $(a_i,\phi_i)_i$, with almost all $a_i,\phi_i$ being zero. The multiplication is defined by
$$
(a_i,\phi_i)_i * (b_i,\psi_i)_i = (a_i b_i,a_{i+1}\psi_i + \phi_i b_i)_i .
$$

Now from the book: 

It is easily seen that the $\hat{A}$-modules can be written in the following way: $M=(M_i, f'_i)_i$, where the $M_i$ are $A$-modules, all but finitely many being zero, the $f'_i$ are $A$-linear maps 
  $$
f'_i \colon DA \bigotimes_A M_i \to M_{i+1} \text{ such that } f'_{i+1}(1 \times f'_i)=0
$$
  for all $i \in \mathbb{Z}$.

I did not understand the ease in seeing the modules that way. Moreover, I do not know why the morphism needs the condition $f'_{i+1}(1 \otimes f'_i)=0$. 
These are my first questions on this subject. Thank you in advance.
 A: For each $j\in\mathbb{Z}$, let $e_j\in\hat{A}$ be the element $(a_i,\phi_i)_i$ with $a_j=1$, $a_i=0$ for $i\neq j$ and $\phi_i=0$ for all $i$. Then the $e_i$ are orthogonal idempotents ($e_i^2=e_i$, $e_ie_j=0$ for $i\neq j$) and $\hat{A}=\sum_ie_i\hat{A}$.
Since $\hat{A}$ does not have a unit, we need to be careful what we mean by a "module". The $e_i$ are kind of "local units", and it makes sense to define a "unital module" $M$ to be one such that $M=\bigoplus_ie_iM$. This is what Happel means by an $\hat{A}$-module. It's easy to see that every module without this restriction is the direct sum of a unital module and a module that is annihilated by $\hat{A}$, $M=(\bigoplus_ie_iM)\oplus N$ where $N=\{m\in M:\hat{A}m=0\}$, so restricting to unital modules doesn't lose much.
Then given a (unital) module $M$, let $M_i=e_iM$. (I think Happel is thinking only of finite-dimensional modules $M$, so $M_i$ will be zero for all but finitely many $i$.) 
If $\theta\in DA$, and $\theta_j=(a_i,\phi_i)_i\in\hat{A}$, where $a_i=0$ for all $i$, $\phi_j=\theta$ and $\phi_i=0$ for $i\neq j$, then $\theta_ie_i=\theta_i=e_{i+1}\theta_i$, so we can define $f_i':DA\otimes_AM_i\to M_{i+1}$ to be the map $\theta\otimes e_im\mapsto \theta_ie_im=e_{i+1}\theta_im$.
Then 
$$f_{i+1}'(1\otimes f_i')(\theta\otimes\psi\otimes e_im)=f_{i+1}'(\theta\otimes e_{i+1}\psi_im)=e_{i+2}\theta_{i+1}\psi_im,$$
which is zero since $\theta_{i+1}\psi_i=0$.
