I'm trying to prove the following assertion in Weibel's Homological Algebra page 125, 5.2.8
Given a homology spectral sequence, we see that each $E^{r+1}_{pq}$ is a subquotient of the previous term $E^r_{pq}$. By induction on $r$, we see that there is a nested family of subobjects of $E^a_{pq}$: $$0 =B^a_{pq}\subseteq \cdots \subseteq B^r_{pq} \subseteq B^{r+1}_{pq} \subseteq \cdots \subseteq Z^{r+1}_{pq}\subseteq Z^r_{pq} \subseteq \cdots \subseteq Z^a_{pq} =E^a_{pq}$$ such that $E^r_{pq}\cong Z^r_{pq}/B^r_{pq}$
If we are working in the category of $R$-modules, then this proposition is trivially seen to be a consequence of this diagram
Because if $B^{r+1}_{pq}$ is contained in $E^r_{pq}$, then it has to contain $0$ which represents $B^{r}_{pq}$. Clearly $B^r_{pq}\subseteq Z^r_{pq}$ and $Z^r_{pq}\subseteq Z^{r-1}_{pq}$ o/w we wouldn't have $Z^{r+1}_{pq}\hookrightarrow E^r_{pq}$.
But I can't generalise this reasoning to work in any abelian category. The diagram still holds, but I really don't know how to prove (using the sub-objects) the claim Weibel does. (Clearly I don't want to use the embedding in $R$-Mod)
Can someone provides me some hints on what is the general argument for abelian cats?
ADDENDUM actually I fear that even in the $R$-Mod case, this doesn't make much sense. In fact, if $B^{r+1}_{pq}\hookrightarrow Z^r_{pq}/B^r_{pq}$, then how come $B^r_{pq}\hookrightarrow B^{r+1}_{pq}$? in fact it'd be clearly a submodule of the kernel of the inclusion, but the inclusion is injective and hence it'd be trivial but in general $B^r_{pq}\neq 0$. So what does Weibel mean here?
The object $B^{r+1}_{pq}$ is not defined to be the image of $d_{p_1q_1}$. Rather, it is the preimage of $\operatorname{im}(d_{p_1q_1})\subseteq E^r_{pq}$ under the quotient map $\pi:Z^r_{pq}\to E^r_{pq}$. So it is not a subobject of $E^r_{pq}$ but of $Z^r_{pq}$ (and hence, by induction, of $E^a_{pq}$). Similarly, $Z^{r+1}_{pq}$ is the preimage of $\ker(d_{pq})$ under $\pi$. The thing you have to check is then that $Z^{r+1}_{pq}/B^{r+1}_{pq}\cong \ker(d_{pq})/\operatorname{im}(d_{p_1q_1})$. This follows from a general fact about pulling back subobjects in abelian categories that should not be too hard to prove: if you have an object $X$ with subobjects $U\hookrightarrow V\hookrightarrow X$, and an epimorphism $\pi:Y\to X$, then $\pi^{-1}(V)/\pi^{-1}(U)\cong V/U$.