In the Milnor and Moore paper, "On the structure of Hopf algebra" proposition 1.7. Said the following: 

> Blockquote 
>  1. $A$ connected $K$-algebra 
>  2. $N$ a left $A$ module and it is connected $K$-graded module i.e. there is an isomoprhism $\eta_N:K\to N_0$ which result in
> augementation $\varepsilon_N:N\to K$
>  3. $C=K\otimes_A N$, here $K$ is considered as $A$-rightmodule by the action $K\otimes A \to_\cong A \to_{\varepsilon_A} K$
>  4. $\Delta :N \to N\otimes C$ morphism of left $A$ modules. 
>  5. $\pi:N\to C$ canonical epimorphism, let $f:C\to N$ such that $\pi f = id_C$ in this paper they used $C$ the object letter to indicate
> the identity morphism
>  6. $(\varepsilon_N\otimes C)\circ \Delta =f$
>  7. $(N\otimes \varepsilon_C)\circ \Delta = N$
>  8. $A\otimes C \to_{i\otimes C} N\otimes C $ is monomorphism.
> 
> If $\tilde{f}$ to be the composition $$A\otimes C
 \to_{A\otimes f} A \otimes N \to_{\varphi_N} N$$ then it is an isomorophism




The map $i$ is defined as the composition 
$$A\to_{\cong} A\otimes K \to_{A\otimes \eta_N} A\otimes N \to_{\varphi_N} N$$
and $\varphi_N$ the left action of $A$ on $N$

**My first question:** in the process to show that $\tilde{f}$ is monomorphism, they showed $\Delta \tilde{f}$ is monomorphism.
 
  They define a filteration on $$F_p(A\otimes C) = \sum_{q\geq p} A\otimes C_q \text{ and } F_p(N\otimes C) = \sum_{q\geq p} N\otimes C_q$$
Then they said: "let $E^0 (A\otimes C)$ the associated bigraded module"
 is this the grading associated with the filteration i.e. $gr_p(A\otimes C) = F_p/F_{p-1}$ ? or it is just the grading on $A$ and $C$ as $K$-modules? I guess it is the second since they said $E^0_{p,q} (A\otimes C) = A_p \otimes C_q$ 
After that they said we identify $E^0(A\otimes C)$ with $A\otimes C$ ? then why did they introduce the $E^0$?

**Second question:** How come the two following morphisms are the same: 
$$A\otimes C \to_{A\otimes f} A\otimes N \to_{\varphi_N} N\to_{\Delta} N\otimes C$$
and $i\otimes C$ which is  
$$A\otimes C \to_{\cong\otimes C } A\otimes K\otimes C  \to_{A\otimes \eta_N\otimes C } A\otimes N \otimes C \to_{\varphi_N} N\otimes C $$

The paper: 
https://t.ly/G1dL