In the Milnor and Moore paper, "[On the structure of Hopf algebras](https://doi.org/10.2307/1970615)" proposition 1.7 said the following: 

>  1. $A$ a connected $K$-algebra.
>  2. $N$ a left $A$ module that is connected as a $K$-graded module i.e. there is an isomorphism $\eta_N:K\to N_0$ which results in an
> augmentation $\varepsilon_N:N\to K$.
>  3. $C=K\otimes_A N$, here $K$ is considered as $A$-rightmodule by the action $K\otimes A \xrightarrow{\cong} A \xrightarrow{\varepsilon_A} K$.
>  4. $\Delta :N \to N\otimes C$ a morphism of left $A$ modules. 
>  5. $\pi:N\to C$ the canonical epimorphism. Let $f:C\to N$ such that $\pi f = \operatorname{id}_C$. In this paper they used $C$ the object letter to indicate
> the identity morphism.
>  6. $(\varepsilon_N\otimes C)\circ \Delta =\pi$.
>  7. $(N\otimes \varepsilon_C)\circ \Delta = N$.
>  8. $A\otimes C \xrightarrow{i\otimes C} N\otimes C $ is a monomorphism.
> 
> If $\tilde{f}$ is the composition $$A\otimes C
 \xrightarrow{A\otimes f} A \otimes N \xrightarrow{\varphi_N} N$$ then $\tilde f$ is an isomorphism.




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

**My first question:** in the process to show that $\tilde{f}$ is a 
 monomorphism, they showed $\Delta \tilde{f}$ is a monomorphism.
 
  They define a filtration on $$F_p(A\otimes C) = \sum_{q\leq p} A\otimes C_q \text{ and } F_p(N\otimes C) = \sum_{q\leq p} N\otimes C_q$$
Then they said: "let $E^0 (A\otimes C)$ be the associated bigraded module."
 Is this the grading associated with the filtration i.e. $\operatorname{gr}_p(A\otimes C) = F_p/F_{p-1}$? 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: 
$$A\otimes C \xrightarrow{A\otimes f} A\otimes N \xrightarrow{\varphi_N} N\to_{\Delta} N\otimes C$$
and $i\otimes C$ which is  
$$A\otimes C \xrightarrow{\cong\otimes C } A\otimes K\otimes C  \xrightarrow{A\otimes \eta_N\otimes C } A\otimes N \otimes C \xrightarrow{\varphi_N} N\otimes C. $$

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